' 


AN 


ELEMENTARY   PHYSICS 


FOR  SECONDARY  SCHOOLS 


BY 


CHARLES  BURTON  THWING,  PH.D.  (BONN) 

PROFESSOR  OF  PHYSICS  IN  KNOX  COLLEGE 

FORMERLY  INSTRUCTOR  IN  PHYSICS  IN  THE  UNIVERSITY  OF  WISCONSIN 
AUTHOR  OF  "EXERCISES  IN  PHYSICAL  MEASUREMENT" 


PART  I:      PRINCIPLES 

PART   II:    LABORATORY   EXERCISES 


ov  TrdAA*  dAAa  TroAv 


BOSTON,  U.S.A. 

BENJ.   H.   SANBORN   &  CO. 
1900 


^5  V,   k 

T^ls 


COPYRIGHT,  1900, 
BY  CHARLES  BURTON  THWING. 


Press  of  Samuel  Usher,  Boston,  Mass. 


PREFACE. 


THAT  there  is  a  demand  for  a  new  School  Physics 
recent  correspondence  with  over  fifteen  hundred  second- 
ary schools  abundantly  shows.  That  there  are  good 
books  now  before  the  educational  public  cannot  be 
denied.  The  good  books  —  those  reasonably  accurate, 
scientific,  and  modern  —  are  too  difficult,  too  diffuse,  or 
require  more  than  the  usual  school  equipment  of  appa- 
ratus. In  the  so-called  attempt  to  "  enrich  the  grammar 
school  courses,"  physics  has  been  in  a  few  cases  intro- 
duced, and  a  moderate  demand  created  for  very  elemen- 
tary books.  Thus  the  instructor  has  been  left  to  a 
choice  between  books  too  elementary  and  simple,  and 
those  that  make  too  great  a  demand  upon  the  pupils  in 
the  usual  secondary  school  course. 

The  author's  experience  as  a  teacher  of  physics,  both 
in  secondary  schools  and  college,  has  given  him  excep- 
tional opportunity  to  know  the  capacity  of  the  average 
student.  His  relations  with  a  large  number  of  teachers 
of  elementary  science  have  been  such  as  to  make  him 
aware  of  the  limitations  under  which  teachers  work, 
both  as  to  specific  preparation  for  the  work  of  teach- 
ing physics  and  still  more  as  to  material  equipment, 

iii 


96805 


IV  PEE  FACE. 

The  book  has  been  written,  therefore,  with  the  needs 
and  limitations  of  the  average  teacher  and  pupil  con- 
stantly in  mind.  Illustrations  are  drawn,  wherever 
possible,  from  facts  already  known  to  the  pupil. 

The  illustrative  experiments  described  in  the  text  are 
few  in  number  and  require  only  simple  apparatus.  For 
teachers  who  have  more  elaborate  apparatus  at  command 
the  accompanying  handbook  gives  full  references  to  text- 
books which  are  in  every  teacher's  library. 

The  plan  of  relegating  to  the  handbook  all  matter 
useful  mainly  to  the  teacher  relieves  the  book  of  much 
fine-print  lumber,  and  allows  the  text  to  proceed  with- 
out interruption  in  a  connected  and  logical  treatment  of 
the  subject. 

The  same  end  is  still  further  subserved  by  putting 
certain  illustrative  experiments  in  the  lists  of  exercises 
for  review  which  follow  each  chapter,  and  by  putting 
the  laboratory  exercises  by  themselves  in  Part  II. 

Flexibility  in  amount  of  work  done  is  attained  by 
giving  a  large  number  of  exercises,  some  of  which  are 
rather  difficult.  Of  the  seventy-five  laboratory  exercises 
in  Part  II,  but  forty  would  be  required  Jor  entrance  to 
Harvard.  Of  the  exercises  in  Part  I,  most  classes 
would  probably  perform  about  half.  The  text  should 
all  be  mastered  by  the  pupil  as  far  as  he  goes. 

The  work  in  Part  II —  which  contains  many  new  and 
practical  ideas  —  should  begin  the  second  week  of  the  term 
and  continue  parallel  with  work  on  the  text.  The  ex- 
periments in  Part  II  are  all  quantitative  and  lead  to 


PEE  FACE.  V 

results  which  may  be  easily  verified  from  data  in  the 
teacher's  hands.  The  tendency  of  the  best  teaching 
practice  is  growing  rapidly  in  the  direction  of  leaving 
qualitative  experiments  to  the  class  room  to  be  per- 
formed by  the  teacher,  or  under  his  direction,  and  giving 
to  the  pupil  experiments  involving  the  law  which  he 
has  already  learned  from  the  book.  To  calculate  the 
breaking  stress  of  one  or  two  particular  wires  from 
measurements  of  the  diameters  of  the  wires  and  the 
force  required  to  break  them,  and  to  find  that  the  value 
obtained  compares  fairly  well  with  the  results  obtained 
by  others  as  given  in  the  book,  is  good  proof  of  the 
correctness  of  the  law.  Careful  weighing  with  the  steel- 
yards and  balance  illustrates  the  law  of  moments  per- 
fectly, while  giving  the  student  a  much  better  training 
in  accuracy  than  he  will  get  by  using  levers  supported 
below  their  centres  of  gravity,  as  suggested  in  many 
books.  Most  of  the  experiments  in  Part  II  have  been 
tested  during  the  past  three  years  in  a  number  of 
schools,  including  both  the  large  city  high  school 
and  the  small  country  school,  with  most  gratifying  re- 
sults. 

The  order  of  arrangement  of  the  chapters  in  Part  I 
demands  a  word  of  explanation.  The  author's  observa- 
tion has  been  that  the  student  encounters  so  many 
difficulties  in  the  early  part  of  the  study  of  mechanics 
that  he  is  likely  to  become  discouraged  before  his  in- 
terest is  fully  aroused  in  the  subject.  He  is  hardly 
familiar  with  the  idea  of  force  before  the  more  complex 


VI  PREFACE. 

idea  of  energy  is  thrust  at  him  and  he  becomes  con- 
fused. Moreover,  he  is  expected  to  comprehend  the 
law  of  conservation  of  energy  when  he  has  studied  but 
one  form  of  energy. 

By  first  treating  all  forms  of  energy  as  forms  of 
motion,  or  capable  of  being  converted  into  motion,  the 
subject  of  energy  is  deferred  until  the  student  is  in 
possession  of  a  large  body  of  facts  which  bear  directly 
upon  it.  Moreover,  the  importance  of  a  thorough  knowl- 
edge of  mechanics  is  emphasized  by  bringing  it  directly 
into  relation  with  heat  and  electricity. 

It  is  believed  that  the  treatment  of  wave  motion  here 
given  is  more  connected  and  clear  than  is  usually  found 
in  elementary  text-books.  The  subject  of  index  of 
refraction  is  treated  with  scientific  accuracy  without  the 
introduction,  either  implicitly  or  by  name,  of  the  trigo- 
nometric functions  (pages  204  and  351).  It  is  the 
author's  firm  conviction  that  treatments  involving  any 
knowledge  of  mathematical  notions  beyond  those  gained 
in  elementary  algebra  and  plane  geometry  are  out  of 
place  in  a  text-book  designed  for  secondary  schools. 
The  definition  of  index  of  refraction  as  a  ratio  of  the 
velocities  of  light  in  the  two  media  calls  attention  to 
the  physical  fact  involved,  while  its  definition  as  the 
ratio  of  the  sines  of  two  angles  directs  attention  to  one 
of  the  methods  usually  employed  in  its  determination. 

Teachers  are  requested  in  examining  the  book  to  read 
the  corresponding  portions  of  the  handbook  in  connec- 
tion with  the  text. 


PREFACE.  vil 

Corrections  or  suggestions  from  teachers  using  the 
book  will  be  gratefully  received  by  the  author,  who 
realizes  that  the  task  of  writing  a  book  which  shall  be 
at  once  simple  and  accurate  is  by  no  means  an  easy 
one. 

I  am  under  obligations  to  Mr.  Frank  Wenner,  In- 
structor in  Knox  Academy,  to  Professor  F.  L.  Bishop, 
of  Bradley  Polytechnic  Institute,  to  Mr.  Allan  C. 
Rearick,  Instructor  in  Science  in  Kewanee  High  School, 
and  to  Mr.  L.  E.  Flanegin,  Superintendent  of  Schools, 
Elm  wood,  111.,  who  have  rendered  valuable  assistance 
in  the  preparation  of  the  book,  either  in  the  way  of 
suggestions  or  in  reading  the  proof,  and  to  Mr.  Wenner 
and  Mr.  G.  A.  McMaster  for  work  upon  the  draw- 
ings. C.  B.  T. 

GALESBURG,  ILL.,  January,  1900. 


CONTENTS.* 


PART  I.  — PRINCIPLES. 


INTRODUCTION. 

The  Place  of  Physics  (a)  among  the  Sciences,  1;  (b)  in  Educa- 
tion, 3;  (c)  in  Practical  Life,  3. 


CHAPTER  I. 
MATTER  AND  MOTION. 

Definitions,  4;  Kinds  of  Motion,  5;  Force,  8;  Velocity,  Accel- 
eration, 10;  Mass,  Momentum,  11;  Newton's  Laws  of  Motion,  13; 
Composition  and  Resolution  of  Forces,  14;  Measurement,  Units, 
20;  Exercises,  23. 

CHAPTER  II. 
BALANCING  FORCES. 
Kinds  of  Equilibrium,  25;  Moment  of  Force,  30;  Gravitation,  32. 

SOME  PROPERTIES  OP  MATTER. 

Elasticity,  Molecules,  Cohesion,  34,  35 ;  Liquids  in  Open  Vessels, 
38;  Capillarity,  41 ;  Floating  Bodies,  46 ;  Liquids  in  Closed  Vessels, 
47;  Gases  in  Open  Vessels,  48;  Pumps,  51-54;  Fluids  in  Contact, 
Diffusion,  55;  Exercises,  59. 

*  The  references  are  to  pages. 

ix 


X  CONTENTS. 

CHAPTER  III. 
HEAT. 

Nature  of  Heat,  61;  Effects  of  Heat,  62;  Temperature,  63;  Ex- 
pansion, 64;  Thermometry,  65;  Quantity  of  Heat,  Specific  Heat, 
70;  Change  of  State,  71;  Latent  Heat,  71;  Conduction,  Convection, 
76-77;  Heat  and  Human  Life,  Weather,  Clothing,  77-79;  Exer- 
cises, 82. 

CHAPTER  IV. 

ELECTRICITY  AND  MAGNETISM. 

Introductory  Remarks,  84;  Magnets,  85;  Magnetic  Forces,  86; 
The  Magnetic  Field,  88;  Terrestrial  Magnetism,  94. 

Electric  Charges,  94;  Induction,  98;  Nature  of  the  Charge,  100; 
Electric  Discharge,  104;  Electrical  Machines,  107;  Capacity,  Con- 
densers, 112;  Electroscope,  114;  Discharge  in  Gases,  116;  Exer- 
cises, 117. 

Electric  Currents,  120;  Magnetic  Effects  of  Current,  122;  Bat- 
teries, 123;  Electrolysis,  126;  Heat  Effects,  Resistance,  128;  Ohm's 
Law,  Units,  130;  Useful  Heat  Effects,  131;  Motion  from  Current, 
135;  The  Telegraph,  137;  Motors  and  Dynamos,  140;  Induction 
Coils,  144;  Exercises,  151. 


CHAPTER  V. 
WORK.    ENERGY.    MACHINES. 

Definitions,  157;  Conservation  of  Energy,  160;  Units,  Equiva- 
lents, 162;  Energy  and  Life,  166;  Machines,  169;  Heat  Engines, 
180;  Storage  and  Transmission  of  Energy,  186;  Exercises,  189. 

CHAPTER  VI. 
VIBRATIONS.    WAVES. 

Vibratory  Motion,  192;  The  Pendulum,  194;  Wave  Motion,  197; 
Reflection,  200;  Refraction,  204;  Exercises,  205. 


CONTENTS.  Xi 

CHAPTER  VII. 
SOUND. 

Nature  of  Sound,  207;  Rods,  Plates,  Bells,  208-209;  Pitch,  210; 
Musical  Intervals,  Consonance,  211;  Resonance,  Pipes,  214;  Veloc- 
ity of  Sound,  216;  Musical  Scales,  220;  Exercises,  224. 

CHAPTER  VIII. 
LIGHT. 

The  Sensation  of  Light,  226;  Shadows,  227;  Images  (a)  by  Small 
Openings,  228;  (ft)  by  Reflection,  229;  (c)  by  Refraction,  235;  Op- 
tical Instruments,  239;  Color  by  Refraction,  The  Spectrum,  243; 
Color  Mixture,  245;  Color  by  Interference,  249;  The  Spectroscope, 
251;  Intensity  of  Light,  255;  Light  and  Electricity,  256;  Space 
Telegraphy,  257;  New  Forms  of  Radiation,  258;  The  Role  of  Wave 
Motion  in  Nature,  259;  Exercises,  260. 

PART  II.  — LABORATORY   EXERCISES. 

INTRODUCTION. 

Measurement  as  a  Form  of  Training,  265;  Sources  of  Error,  266; 
Hints  to  the  Student,  267. 

CHAPTER  IX. 
LENGTH. 

Units  and  Equivalents,  268;  The  Metre  Stick,  269;  Dividers  and 
Calipers,  273;  The  Vernier,  278;  Eye  Estimations,  282;  The  Mi- 
crometer, 283;  The  Spherometer,  285. 

CHAPTER  X. 
MASS. 

Definitions  and  Units,  288;  The  Balance,  289;  Weighing  by 
Swings,  297;  Density  (a)  by  Dimensions  and  Weight,  300;  (6)  by 
Displacement,  302;  (c)  by  Archimedes'  Principle,  304;  (d)  with 
the  Bottle,  305;  (e)  by  Balancing  Columns,  307;  Tables  of  Densi- 
ties, 309. 


xii  CONTENTS. 

CHAPTER  XI. 
TIME. 

Units,  310;  The  Pendulum,  311;  Torsional  Pendulum,  312;  De- 
termination of  Gravity,  313;  The  Pulse,  313;  The  Respiration,  314; 
Estimation  of  Time,  314. 

CHAPTER  XII. 
FORCE. 

Pressure  of  the  Air,  316;  Use  of  the  Manometer,  319;  Capil- 
lary Constant,  320;  Modulus  of  Elasticity,  322;  Breaking  Strength, 
326;  Friction,  327. 

CHAPTER  XIII. 
HEAT. 

Thermometry,  329;  Expansion,  332;  Specific  Heat,  333;  Humid- 
ity, 336. 

CHAPTER  XIV. 
ELECTRICITY. 

Resistance,  339;  Wheatstone's  Bridge,  340;  Specific  Resistance, 
342;  Temperature  Coefficient  of  Resistance,  343;  Potential  Differ- 
ence, 345. 

CHAPTER  XV. 

SOUND. 

Velocity  of  Sound  (a)  in  Air,  346;  (6)  in  Solids,  347;  Pitch  of 
a  Fork,  3/9. 

CHAPTER  XVI. 

LIGHT. 

Photometry,  350;  Index  of  Refraction,  351;  Lenses,  354;  Mag- 
nification, 356. 

INDEX  OP  PROPER  NAMES,  357 ;  GENERAL  INDEX,  361. 


PHYSICS. 


PART  I.  — PRINCIPLES. 

INTRODUCTION. 

i.  The  Place  of  Physics  among  the  Sciences.  - 

Natural  Science  has  for  its  aim  the  explanation  of  nat- 
ural phenomena  —  the  things  which  happen  about  us 
every  day.  It  was  originally  all  comprised  under  Nat- 
ural Philosophy,  or  Physics.  Our  knowledge  of  natural 
phenomena  has  so  increased  during  the  last  few  cen- 
turies that  no  one  man  could  hope  to  master  all  of  it. 
Natural  Science  has  been  divided,  therefore,  into  several 
branches.  Astronomy  deals  with  phenomena  wholly 
beyond  our  control.  Chemistry  studies  changes  which 
result  in  the  formation  of  new  substances.  Biology 
deals  with  the  phenomena  manifested  in  living  things. 
Physics  is  now  confined  to  the  study  of  those  changes 
which  occur  in  the  position,  appearance,  or  state  (solid, 
liquid,  gaseous)  of  bodies  ;  it  seeks  to  trace  each  change 
to  an  adequate  cause  and  to  discover  the  laws^ which 
govern  the  phenomena  observed.  Physics  includes  the 
subjects  of  mechanics,  heat,  electricity,  sound,  and  light. 
The  laws  enunciated  in  physics  are  assumed,  and  the 
instruments  described  in  physics  are  employed,  in  the 
other  sciences. 


2  INTE  OD  UC  TION. 

Physics,  then,  is  at  the  foundation  of  all  the  sciences. 
Physics  itself  is  founded  upon  a  body  of  observed  facts 
which  are  interpreted  by  reason  in  accordance  with  the 
principles  of  mathematics.  The  exact  expression  of 
the  relation  between  a  large  number  of  similar  facts  is 
called  a  law.  Men  had  observed  for  ages  that  bodies 
unsupported  in  the  air  tend  to  fall  toward  the  earth. 
It  was  only  after  many  years  of  patient  observation  and 
the  making  of  careful  measurements  that  the  fact  be- 
came known  that  all  bodies  at  the  surface  of  the  earth 
fall,  if  unimpeded,  with  the  same  velocity,  the  velocity 
gained  being  32.16  feet  (9.8  metres)  for  every  second 
of  time  the  body  is  falling.  This  fact,  after  being  tested 
by  innumerable  experiments,  is  now  known  as  the  law  of 
falling  bodies. 

Among  the  many  new  facts  which  are  constantly  being 
discovered  there  are  always  some,  the  relation  of  which 
to  the  body  of  known  facts  is  not  yet  apparent.  When 
a  large  number  of  such  facts  have  been  collected  some 
bold  philosopher  offers  an  explanation,  or  as  scientists 
say,  a  hypothesis.  This  hypothesis,  if  it  seems  to  other 
scientists  to  be  worthy  of  consideration,  is  tested  to  see 
if  it  will  explain  all  the  known  facts.  If  it  does  explain 
the  known  facts,  and  especially  if  it  leads  to  the  dis- 
covery of  new  facts,  it  comes  in  time  to  be  accepted  as 
a  theory.  The  Atomic  Theory,  for  example,  attempts  to 
harmonize  all  the  known  facts  of  chemistry,  and  does  so 
to  a  degree  that  was  hardly  anticipated  by  those  who 
first  proposed  it.  The  theory  seldom  has  so  sure  a  f  oun- 


INTRODUCTION.  3 

elation  as  the  separate  laws  or  facts  which  it  attempts  to 
explain,  yet  it  serves  a  most  useful  purpose,  even  when 
it  does  no  more  than  aid  us  in  keeping  those  laws  and 
facts  in  memory. 

2.  The  Place  of  Physics  in  Education.  —  The  study 
of  physics  involves  to  an  exceptional  degree  the  train- 
ing of  the  senses  to  perceive,  the  hand  to  perform,  and 
the  mind  to  interpret,  on  the  one  hand,  what  the  senses 
observe ;  to  direct,  on  the  other,  what  the  trained  hand 
shall  perform.     In  cultivating  accuracy,  it  discourages 
slovenliness  and  untruthfulness  and  stimulates  the  mind 
to  that  highest  of  human  pursuits,  the  search  for  truth. 

3.  The  Place  of  Physics  in  Practical  Life.  —  Me- 
chanical inventions  are  not,  as  many  thoughtless  people 
suppose,  mere  lucky  accidents ;  they  are  the  direct  result 
of  a  knowledge  of  the  laws  of  nature.     The  man  who 
discovers  a  new  law  of  physics  seldom  sees  how  his  dis- 
covery is  to  be   of  any  practical  benefit  to  mankind. 
Faraday,  toiling  patiently  in  his  laboratory  to  discover 
the  laws  of  electro-magnetic  induction,  never  dreamed 
that  liis  discovery  of  these  laws  would  one  day  make 
possible  the  electric  lighting  of  our  homes  and  streets, 
the  electric  propulsion  of  railway  trains,  and  the  trans- 
mission of  spoken  words  across  a  continent.     Yet  these 
are  but  a  few  of  the  applications  of  Faraday's  discov- 
eries to  the  daily  life  of  men. 

Enough  has  been  said,  it  would  seem,  to  make  it  evi- 
dent that  there  are  excellent  reasons  for  giving  physics 
a  place  in  the  course  of  study. 


CHAPTER   I. 
MATTER  AND   MOTION. 

4.  Matter  and  Motion  Defined.  —  We  have  seei- 
already  that  physics  is  the  study  of  the  changes  that 
occur  in  the  things  about  us.  We  shall  see  as  we  pro- 
ceed that  all  changes  in  the  appearance  of  things  are 
due  to  motion,  either  of  the  things  themselves  or  of 
their  parts. 

Motion  may  be  denned  as  change  of  position,  the 
position  of  an  object  being  determined  by  its  distance 
and  direction  from  other  objects.  In  geometry  we  deal 
with  motions  of  points  and  lines,  as  when  a  point  is  said 
to  move  so  that  its  path  is  a  straight  line  or  a  circle. 
All  physical  changes,  however,  imply  the  motion  of 
matter. 

Matter  is  a  term  with  the  meaning  of  which  we  are 
all  more  or  less  familiar,  yet  a  term  which  it  is  not  easy 
to  define.  We  know  matter  primarily  by  the  effort 
required  to  move  it  out  of  its  place.  If  we  walk  in  air 
we  are  so  accustomed  to  the  resistance  which  it  offers 
to  our  movements  that  we  think  of  our  motion  as  un- 
impeded ;  but  if  we  walk  in  water  or  through  thick 
underbrush  we  are  conscious  of  a  resistance  which  can 
be  overcome  only  by  a  considerable  effort  on  our  part. 
Matter  also  makes  itself  known  to  us  through  the  senses 

* 


KINDN    OF  MOTION.  5 

of  sight,  hearing,  smell,  and  taste.  It  is  chiefly  through 
the  senses  of  sight  and  hearing,  especially  the  former, 
that  we  gain  our  knowledge  of  physical  changes. 

A  definite  portion  of  matter,  separated  from  other 
matter,  is  called  a  body.  The  various  kinds  of  matter 
of  which  bodies  are  composed  are  called  substances. 
Thus  a  pencil  is  a  body,  composed  of  wood  and  graphite. 
Substances  possess  certain  characteristic  properties : 
cedar  wood  is  of  a  reddish  color,  is  brittle,  and  will 
float  in  water ;  graphite  is  black,  leaves  a  trace  on  paper, 
and  is  heavier  than  water.  The  pencil  possesses  in  its 
different  parts  the  properties  of  the  substances  which 
compose  it,  and  has  besides  a  definite  form  and  size. 

The  amount  of  matter  in  a  body  is  called  the  mass  of 
the  body.  The  amount  of  matter  in  unit  volume  of  a 
substance  is  called  the  density  of  the  substance. 

The  various  properties  of  bodies  are  made  manifest 
to  us  by  the  motions  which  are  transmitted  from  these 
bodies  to  our  organs  of  sense.  We  shall  best  explain 
the  properties  of  matter,  therefore,  by  examining  the 
motions  of  bodies.  Meanwhile  we  must  call  to  mind 
frequently  many  facts  which  have  already  come  to  our 
notice  and  keep  ourselves  ever  alert  to  see  new  facts 
which  may  serve  to  illustrate  the  subject  in  hand. 

5.  Kinds  of  Motion. — I.  A  body  may  move  from 
one  position  to  another  in  such  a  way  that  the  direction 
from  one  point  in  the  body  to  another  point  in  the  body 
remains  unchanged.  Such  motion  is  called  motion  of 


6 


MATTER   AND   MOTION. 


translation,  or  translator^  motion.     Thus  the  cube  CDB 
(Fig.  1)  may  have  moved   to   the  position   O'D'B'  in 


/ 
/ 

/ 

/ 

/ 
/ 
/ 
/ 

/ 

/ 

/ 
/ 

/ 
/ 

A 

'     , 

I      ' 
^/ 


such  a  way  that  CD 
did  not  depart  from 
the  vertical  direction. 
The  motion  was  then 
translatory.      If    the 
successive     positions 
of  a  point,  as   (7,  lie 
in  a  straight   line,  the  motion  is  recti- 
linear; if  in  a  curved  line,  curvilinear. 
B  Thus,   if    C  followed    the  straight  line 
CFC1,  the  motion  was  rectilinear ;  if  the 
line  CEC',  curvilinear ;  and  if   CEC'  is 
nl  I     a    segment  of   a  cir-  ^, 

FIG.  l.  c}e^  ft  wag  circular. 

II.  A  body  may  move  in  such  a 
way  that  the  distance  of  every  point 
in  it  from  a  fixed  point  remains  un- 
changed, while  the  direction  from  one 
point  in  the  body  to  another  point  in 
the  body  changes.  Thus  the  cube  in 
Fig.  2  may  move  from  the  position  £DB  to  <7'2XJ5',  the 


o- 


D 

>B 


Y 

FIG.  2. 


KINDS   OF  MOTION.  7 

point  0  remaining  at  a  fixed  distance  from  0,  while  the 
direction  of  CD  is  constantly  changing.  Such  motion  is 
rotary.  A  line  drawn  through  0  such  that  all  points 
on  the  line  remain  on  the  line  during  rotation  is  called 
the  axis  of  rotation.  The  path  of  every  particle  in 
the  body  is  a  circle,  having  its  centre  on  the  axis. 

III.  A  body  may  move  in   such   a  way  that  after 
moving  a  certain  distance  in  one  direction  it  stops  and 
retraces  its  path  in  the  opposite  direction,  as  the  pendu- 
lum of  a  clock  does.     Such  motion  is  called  motion  of 
vibration,  or  vibratory  motion.     Vibratory  motion  is  often 
so  rapid  that  the  eye  cannot  follow  it.     The  prongs  of 
a  tuning  fork,  the  string  of  a  violin,  indeed,  every  body 
which  is  producing  sound  vibrates  thus  rapidly. 

IV.  When  a  vibrating  body  is  surrounded  by  a  sub- 
stance the  particles  of  which  may  be  set  in  vibration, 
the  particles  nearest  the  vibrating  body  will  be  set  in 
vibration  first.     These  particles  will  communicate  their 
motion  to  those  next  them,  and    the  disturbance  will 
thus   spread   throughout  the  substance.     Such   motion 
is   called   wave    motion.      It  is  usually  represented  by 

a  wavy  line, 
as  in  Fig.  3. 

FIG.  3.  *  The   path  of 

any  particle  is  not  such  a  wavy  line,  but  is  usually  a 
short  straight  line  or  a  small  ellipse.  No  single  particle 
moves  forward,  but  each  particle  vibrates  for  a  short 
time,  sets  its  neighbor  in  vibration,  and  then  comes  to 
rest.  The  motion  progresses  from  particle  to  particle, 


8  MATTER   AND   MOTION. 

each  particle  vibrating  in  turn.  A  water  wave  moves 
across  the  lake  :  a  chip  on  the  water  moves  up  and  down, 
but  not  forward.  It  is  by  wave  motion  that  sound  and 
light  reach  us  from  distant  objects.  The  particles  of  the 
bodies  themselves,  and  of  the  intervening  medium  as 
well,  vibrate  in  very  short  paths ;  the  sound  or  light  — 
that  is,  the  wave  motion  —  travels  to  great  distances. 

6.  The  Cause  of  Motion,  Force.  —  It  is  the  uni- 
versal experience  of  mankind  that  bodies  not  endowed 
with  life  have  no  power  to  put  themselves  in  motion. 
Newton  perceived  also,  what  is  equally  true,  that  bodies 
once  in  motion  have  no  power  to  stop  themselves  or 
change  their  direction  of  motion.  If,  then,  we  see  a 
body  which  was  at  rest  begin  to  move,  or  a  body  which 
was  rising  in  the  air  begin  to  fall,  we  say  something 
has  acted  upon  it.  Provisionally,  while  we  are  seeking 
to  find  what  that  something  may  be,  it  is  convenient  to 
give  it  a  name.  We  call  by  the  name  force  any  cause 
which  tends  to  change  the  motion  of  bodies.  If  a 
body,  as  a  ship  or  a  railway  train,  starts  from  the  state 
of  rest  (the  condition  in  which  it  has  no  motion  rela- 
tive to  bodies  near  it),  and  moves  due  east,  we  say  a 
force  has  acted  upon  it  from  the  west.  If  it  goes 
faster  and  faster  toward  the  east  we  say  a  force  con- 
tinues to  act  from  the  west.  If  it  goes  more  slowly 
eastward  we  say  a  force  is  acting  from  the  east, 
which,  if  it  continues  to  act,  will  bring  the  body  to 
rest.  If  a  stone  is  thrown  horizontally  it  will  not 


THE  NATURE   OF  FORCE*  '.-      ^  •'•"'  9 


continue  to  move  in  a  horizontal  path,  but  will  gradually 
approach  the  ground.  If  we  saw  a  boy  throw  the  stone 
we  say  we  know  the  cause  of  the  horizontal  motion. 
We  saw  nothing  pushing  or  pulling  the  stone  toward 
the  ground,  yet  because  it  went  downward  we  say  there 
was  a  force  acting  in  the  downward  direction.  The 
fact  that  all  bodies  tend  to  move  earthward  has  led  us 
to  give  to  that  force  which  causes  such  motion  toward 
the  earth  a  special  name,  gravity.* 

7.  The  Nature  of  Force.  —  As  was  suggested  in  the 
last  paragraph,  there  are  forces  the  nature  of  which  is 
evident,  and  other  forces  the  nature  of  which  is  not 
known.  It  may  be  said,  however,  that  so  far  as  we 
know  the  nature  of  force,  every  force  is  itself  the  result 
of  motion.  The  wind  is  able  to  move  the  ship  because 
the  air  is  itself  in  motion.  The  steam  in  the  cylinders 
of  the  locomotive  is  able  to  drive  the  train  because  mo- 
tion in  the  form  of  heat  was  first  imparted  to  the  steam 
in  the  boiler. 

The  manner  in  which  the  motion  of  an  engine  when 
transmitted  by  belting  to  a  dynamo-machine  is  so 
changed  that  it  may  be  carried  along  a  slender  wire  for 
miles  to  be  again  converted  into  visible  motion  in  the 
street  car  is  not  yet  well  understood,  but  it  seems  prob- 
able that  motion  is  present  during  every  step  of  the 
process  of  transferring  motion  from  the  engine  to  the 
car,  even  though  we  are  not  able  to  perceive  it. 

*  Gravity  from  Latin  gravis,  heavy,  gravitas,  weight. 


10  MATTE  It  AND  MOTION. 

So  far  as  we  know,  every  bird  came  from  an  egg, 
which  egg,  in  turn,  had  a  bird  for  its  mother ;  so  far  as 
we  know,  every  motion  is  the  result  of  some  antecedent 
motion. 

8.  Rate  of  Motion.     Speed.     Velocity.  —  We  have 
all  observed  that  bodies  in  motion    do    not   all    move 
equally  fast.    A  strong  wind  Avill  drive  a  windmill  faster 
than  a  light  breeze.     A  professional  pitcher  can  throw 
a  ball  swifter  than  a  schoolboy  can.     When   distance 
travelled,  only,  and  not  direction  of  motion  is  considered, 
we  call  the  distance  travelled  in  one  second  the  speed  of 
a  body.      Velocity  is  rate  of  change  of  position,  and  in- 
cludes both  speed  and  direction  of  motion.     If  the  dis- 
tinction between  speed  and  velocity  is  carefully  made 
much  confusion  will  be  avoided.     A  train  which  travels 
240  miles  in  four  hours  has  an  average  speed  of  60 
miles  per  hour,  or  one  mile  per  minute,  or  88  feet  per 
second. 

length  of  path  traversed 
Speed  =  — -  — ,    or 

time  occupied 

>=4- 

t 

9.  Change    of     Velocity.      Acceleration.  —  Motion 
with  constant  speed  and  in  a  constant  direction  is  con- 
stant motion.     Motion  which  changes  either  in  speed  or 
direction  is  called  accelerated  motion,  and  the  rate  of 


QUANTITY   OF  MOTION.  11 

change  of  motion  of  a  body  is  called  its  acceleration. 
A  body  which  moves  five  centimetres  in  the  first  second, 
ten  centimetres  in  the  second,  fifteen  in  the  third, 
etc.,  has  a  constant  acceleration  of  five  centimetres 
per  second.  A  body  moving  in  a  circle  with  uniform 
speed  is  an  example  of  accelerated  motion  ;  for,  though 
the  speed  remains  constant,  the  direction  of  motion  is 
constantly  changing.  It  takes  a  force  constantly  acting 
to  produce  a  constant  change  of  speed  in  the  first  case. 
It  takes  a  force  constantly  acting  to  produce  the  con- 
stant change  of  direction  in  the  second  case.  If  a 
heavy  body  be  whirled  by  a  string  held  in  the  hand  we 
are  conscious  of  a  constant  pull  which  we  must  exert 
upon  the  string  to  keep  the  ball  from  moving  off  in  a 
straight  line. 

We  may  now  define  force  as  any  cause  capable  of 
producing  acceleration. 

To  sum  up  : 

change  of  velocity 

Acceleration  =  -  -  ;  —  '         —  -  —  —  -,  or 

time  in  which  change  occurred 


10.  Quantity  of  Motion.  Mass.  Momentum.  - 
A  given  force  will  produce  a  greater  change  of  velocity 
in  a  light  body  than  in  a  heavy  one.  The  amount  of 
acceleration  alone  is  not  a  measure  of  the  magnitude 
of  a  force  :  we  must  also  take  into  account  the  amount 
of  matter  moved.  The  quantity  of  matter  composing 


12  MATTE  11  AND   MOTION. 

a  body  is  called  the  mass  of  that  body.  The  mass  of  a 
body  is  measured  by  the  acceleration  a  given  force  will 
produce  upon  the  body.  The  greater  the  mass  the 
smaller  the  acceleration  produced  by  a  given  force. 
We  are  accustomed  to  form  a  rough  estimate  of  the 
force  we  ourselves  exert  by  the  effort  we  are  conscious 
of  putting  forth.  We  are  conscious  of  more  effort  when 
we  push  aside  a  brick  with  the  foot  than  when  we  push 
aside  a  block  of  Avood  of  the  same  size.  We  say  the 
brick  is  harder  to  move  than  the  block,  and  therefore 
has  more  matter  in  it. 

Quantity  of  motion  is  called  momentum.  It  is  equal 
in  amount  to  the  product  of  the  numbers  representing 
the  mass  and  velocity  of  the  body. 

Momentum  =.  mass  X  velocity,  or 

M=mv 


Since  the  change  of  momentum  of  a  body  is  the  meas- 
ure of  the  force  applied  to  it,  and  since  the  mass  of  the 
body  is  not  supposed,  in  our  discussion  of  motion,  to 
suffer  any  change  in  amount,  it  follows  that  the  force 
expended  in  producing  any  given  motion  is  measured 
by  the  product  of  the  number  of  units  of  mass  times 
the  number  of  units  of  acceleration. 

Force  =  mass  X  acceleration,  or 
CO    f=vna 

With  these  preliminary  definitions  in  mind,  we  should 
now  be  prepared  to  state  and  discuss  the  three  famous 


NEWTON'S  LAWS  OF  MOTION.  13 

axioms  of  Sir  Isaac  Newton,  which  are  known  as  New- 
ton's Laws  of  Motion. 

n.  Newton's  Laws  of  Motion.  —  I.  Every  body 
continues  in  its  state  of  rest  or  uniform  motion  in  a 
straight  line  unless  acted  upon  by  some  external  force. 

11.  Amount  of   motion    is  proportional  to  the  force 
applied  and  in  the  direction  in  which  the  force  acts. 

III.  Action  and  reaction  are  equal  in  amount  and 
opposite  in  direction. 

12.  Discussion  of  the  First  Law.  —  We  have  seen 
in  Section  6  that  force  is  required  to  change  motion, 
that  is,  to  alter  the  speed  or  direction  of  a  body.     If 
the  body  is  at  rest  its  speed  is  zero,  and,  since  it  has  no 
motion,  it  has  no  direction  of  motion.     If  no  force  is 
applied  the  body  remains  at  rest. 

When  a  body  is  in  motion  a  force  would  be  required 
to  produce  any  acceleration  of  that  motion,  that  is,  to 
change  either  the  speed  or  direction  of  the  motion.  It 
follows  that  the  body  continues  in  its  present  state  un- 
less a  force  is  applied.  The  helplessness  of  matter  is 
called  inertia,  and  force  is  sometimes  said  to  be  spent 
in  overcoming  inertia.  Since,  however,  inertia  is  a 
purely  negative  property  of  matter  it  seems  better  to 
say  that  force  is  spent  in  producing  change  of  motion 
than  to  say  that  force  overcomes  the  tendency  of  bodies 
not  to  move. 

Illustrations  of  the  first  law  will  occur  to  every  one. 


14  MATTER  AND  MOTION. 

A  person  who  jumps  from  a  moving  car  keeps  the  mo- 
tion of  the  car  till  his  feet  strike  the  ground.  His  head 
continues  to  move  forward  even  then,  and  unless  he 
runs  forward  it  is  quite  probable  that  his  head,  too,  will 
strike  the  ground  before  it  loses  all  its  motion.  Drops 
of  water  fly  off  from  a  rapidly  rotating  grindstone. 
Bits  of  niud  are  thrown  from  a  carriage  wheel.  Kail- 
ways  have  the  outer  rail  laid  higher  than  the  inner 
where  sharp  curves  occur  to  prevent  the  engine  from 
leaving  the  track,  as  it  would  do  if  it  kept  on  in  a 
straight  line.  The  rotation  of  the  earth  causes  the 
water  of  the  ocean  to  heap  up  at  the  equator  until  it  is 
about  thirteen  miles  higher  there  than  at  the  poles. 
The  Mississippi  River,  therefore,  notwithstanding  its 
source  is  1,575  feet  above  sea  level,  is  really  about  two 
miles  higher  at  its  mouth  than  at  its  source.  If  the 
earth  should  cease  to  rotate  the  Mississippi  would  flow 
rather  swiftly  toward  the  north. 

13.  Discussion  of  the  Second  Law.  Composition 
and  Resolution  of  Forces.  —  In  Section  10  we  have 
defined  "  amount  of  motion,"  or  momentum,  as  mass 
times  velocity.  The  fact  that  the  velocity  of  a  body 
has  changed  is  proof  that  a  force  has  acted  upon  it. 
The  second  law  implies  that  no  matter  how  many  forces 
are  acting  upon  a  given  body  at  one  time  each  force 
will  produce  the  same  effect  as  if  it  alone  were  acting, 
and  the  total  effect  must  be  found  by  combining  the 
effects  of  all  the  forces.  This  may  be  done  in  a  very 


DISCUMICW   OF  THE  SECOND  LAW.  15 

simple  way  when  the  magnitude  and  direction  of  each 
of  the  forces  are  knoAvn.  A  line  has  both  length  and 
direction.  A  line  may  be  used,  therefore,  to  represent 
the  magnitude  and  direction  of  a  force.  Let  the  mag- 
nitude of  each  force  be  represented  by  the  length,  and 
the  direction  of  each  force  by  the  direction,  of  a  line. 
The  absolute  length  of  the  lines  is  of  no  importance  so 
long  as  their  relative  lengths  correspond  to  the  relative 
magnitude  of  the  forces.  Suppose  it  is  required  to  find 
the  magnitude  and  direction  of  the  force  which  is  the 
equivalent  of  two  forces,  one  a  force  of  magnitude,  4, 
acting  from  the  west,  the  other  a  force  of  magnitude,  3, 
acting  from  the  south,  as  ^ 
if  A  should  kick  a  foot- 
ball from  the  south  with 
a  force  of  three  pounds,  T 

while  at  the  same  instant 

iVl  _^_____^_^________ 

B   kicks  it  from  the  west  _> 

with  a  force  of  four  pounds.  FIG.  4. 

Represent  the  force  of  3  from  the  south  by  the  line 
MN  (Fig.  4)  and  the  force  of  4  from  the  west  by  the 
line  MP.  Both  forces  are  supposed  to  be  applied  at 
M.  The  arrowheads  indicate  the  direction  of  motion. 

Now,  since  each  force  produces  the  same  effect  as  if 
it  alone  were  acting,  we  may  suppose  the  ball  to  move 
long  enough  under  the  influence  of  the  first  force  alone 
to  reach  the  point  N,  then  under  the  influence  of  the 
second  force  only  to  move  toward  the  west  from  N  for 
a  like  length  of  time.  The  ball  would  arrive  in  the 


16  MATTER  AND  MOTION. 

time  specified  at  the  point  R,  which  is  the  same  point  it 
would  reach  if  the  two  motions  occurred  at  the  same 
instant.  The  ball  has  now  moved  from  M  to  R,  hence 
the  line  MR  represents  by  its  length  and  direction  the 
magnitude  and  direction  of  a  force  which  is  equivalent 

^__        to  the  two  forces  MN  and 

Nl~"  4  ,'''^     MP   acting    at   M.     This 

xxx  equivalent  force   is   called 

xx  the  resultant  of  the  other 

xx  two  forces.     Its  numerical 


value  is  readily  found  from 
FIG.  5. 

geometry : 


M 


When  the  directions  of  the  forces  make  any  other 
than  a  right  angle  with  each  other  the  value  of  the 
resultant  may  be  found  by  constructing  a  good-sized 
diagram  to  scale  and  measuring  the  length  of  the  line 
which  represents  the  resultant. 
If  more 
than  two 
forces  are  act- 
•D  ing  at  a  point 
the  lines  rep- 
resenting the 
forces  are 

joined  end  to  FIG' 7' 

end.     The  line    connecting   the    beginning  and  end  of 
the  chain  thus  formed  represents  the  resultant.     Thus 


DISCUSSION'  OF  THE  SECOND  LAW.  17 

forces  AB,    AC,    AD,  AE,    AF,  act  at  the  point    A 
(Fig.  6). 

Draw  AB'   \\  AB 

B'C!  ||  AC 
C'D'  ||  AD 
D'E'  ||  AE 
E'F'  ||  AF 

Then  A'F'  (Fig.  7)  is  the  resultant. 

The  method  employed  for  two  forces  is  commonly 
called  the  triangle  of  forces.  It  is  but  one  case  of  the 
more  general  method  called  the  polygon  of  forces.  If 
the  two  forces  act  in  the  same  straight  line  their  resultant 
is  evidently  their  sum  or  their  difference,  according  to 
whether  they  act  in  the  same  or  in  opposite  directions. 

The  process  of  finding  the  resultant  of  two  or  more 
forces  is  called  the  composition  of  forces. 

Since  the  velocity  produced  upon  a  given  mass  is 
proportional  to  the  force  producing  it,  it  is  evident  that 
velocities  may  be  compounded  exactly  as  forces  are. 

It  is  often  desirable  to  resolve  the  force  acting  upon 
a  body  into  two  or  more  forces.  The  process  is  called 
resolution  of  forces.  It  is  the  converse  of  composition 
of  forces.  A  common  case  is  that  of  a  body  moving 
along  a  path  which  is  not  parallel  to  the  direction  of 
the  force  acting,  as  when  a  ship  sails  westward  by 
means  of  a  south  wind.  A  somewhat  simpler  case  is 
the  following:  A  car  standing  upon  track  1  (Fig.  8)  is 
to  be  pushed  west  by  a  switch  engine  on  track  2  by 
means  of  a  rigid  iron  bar  AB.  According  to  the  second 


18 


MATTER   AND   MOTION. 


law  of  motion,  the  force  which  moves  the  car  west  must 
act  in  that  direction.  Only  a  portion  of  the  force  AB 
acts  westward.  Let  us  resolve  AB,  then,  into  two 
components,  one  westward,  CB,  producing  motion,  one 
northward,  AC,  producing  pressure  on  the  tracks. 


FIG.  8. 

Knowing  the  distance  apart  of  the  tracks  and  the  length 
of  the  bar  AB  we  can  at  once  determine  the  numerical 
value  of  each  of  the  two  components. 

The  reason  is  now  obvious  why  the  resultant  of  sev- 
eral forces  not  in  the  same  straight  line  is  always  less 
than  the  sum  of  the  forces.  It  is  also  evident  that 
when  the  direction  of  motion  is  not  in  the  direction 

of    a    certain   force    there 


v  ^xx         must  be  other  forces  acting 

\  ^xx  at  the  same  time  with  that 

\         x*x  force.     The    direction    of 

XAT  motion  is  always  the  direc- 

M  ''  tion  of  the  resultant  of  all 

the  forces  acting. 

In  the  case  given  on  page  16  we  may  resolve  MN 
(Fig.  9)  into  two  forces,  MP  producing  motion  and 
PN  not  producing  motion.  Similarly  we  may  resolve 
NR  into  PR  producing  motion  and  NP  not  producing 
motion.  The  sum  of  the  forces  producing  motion  is: 


DISCUSSION   OF   THE    THIRD  LAW.  19 


MP  -|-  PR  =  MR.     The  sum  of  the  forces  not  produc- 
ing motion  is  :  NP  +  PN==  0. 

14.  Discussion  of  the  Third  Law.  —  The  third  law 
of  motion  implies  that  there  are  always  two  bodies  con- 
cerned in  every  motion.  If  a  moving  bat  strikes  a  ball 
\vhich  is  moving  in  the  opposite  direction  the  amount  of 
motion  destroyed  in  each  is  equal.  If  a  croquet  ball 
at  rest  is.  struck  by  a  mallet  so  that  the  mallet  conies  to 
rest  the  ball  will  move  forward  with  a  momentum  equal 
to  that  which  the  mallet  had  before  impact.  The  ball 
has  exerted  as  much  force  in  stopping  the  mallet  as  the 
mallet  has  exerted  in  moving  the  ball. 

If  a  man  in  a  large  boat  draw  a  small  boat  toward  him 
in  the  water  by  means  of  a  rope  both  boats  will  move, 
but  the  small  one  will  move  faster,  and,  therefore,  farther, 
than  the  large  one.  Their  relative  velocities  may  readily 
be  found  as  follows  :  Call  *  mv  vv  Ml  and  mv  i>2,  M2 
the  masses,  velocities-,  and  momenta  of  the  two  boats. 
Then  M  =.  mv 


and  since     Ml  = 
mivl  = 
If  we  suppose    ml  =  5 
w2=2 
v.<=  10 

2   x  10 

=4 


g 

or  the  small  boat  will  move  two  and  one  half  times  as 
fast  as  the  large  one. 

*  Read  m,  "  m  sub  one." 


20  MATTER  AND  MOTION. 

MEASUREMENT.      UNITS. 

15.  The  Fundamental  Units.  —  In  our  discussion  of 
motions  and  the  forces  which  produce  them  we  have 
assumed  that  it  is  possible  to  express  in  numbers  the 
mass  of  a  body,  the  distance  it  travels,  and  the  time 
spent  in  motion.  The  exact  determination  of  these 
and  similar  quantities  is  one  of  the  principal  tasks  of 
the  physicist,  for  upon  such  measurements  every  law 
and  theory  in  physics  rests. 

A  moment's  reflection  will  show  that  to  measure 
velocity  we  must  measure  distance  and  time,  to  measure 
momentum  we  must  measure  mass,  distance,  and  time. 
It  has  been  found  possible  to  express  all  physical  quan- 
tities, electrical  and  magnetic  included,  in  terms  of  the 
three  fundamentals,  length,  mass,  and  time. 

For  the  measurement  of  any  quantity  we  require  a 
unit  of  quantity  of  the  same  sort,  the  magnitude  of 
which  has  been  determined  by  cojnparison  with  some 
standard  agreed  upon  by  scientists,  The  system  of 
units  based  upon  the  metre  and  known  as  the  metric 
system  is  in  universal  use  by  scientists  and  is  fast  being 
adopted  by  the  civilized  nations  of  the  world  for  use  in 
commercial  transactions. 

A  metre  is  the  length  of  a  certain  bar  preserved  at 
Paris,  copies  of  which  are  in  the  possession  of  the 
Bureau  of  Weights  and  Measures  at  Washington.  Its 
length  is  equal  to  39.37  English  inches,  or  about  1.4. 
yards.  The  metre  is  divided  into  100  centimetres,  as 
our  dollar  is  divided  into  100  cents.  The  centimetre  is 


DERIVED    UNITS.  21 

divided  into  ten  millimetres.  For  long  distances  the 
kilometre  is  used.  A  kilometre  equals  1,000  metres. 
Tables  of  English  equivalents  will  be  found  in  Part  II. 

The  unit  of  mass  is  the  gram.  A  gram  is  the  mass 
of  one  cubic  centimetre  of  water  at  a  certain  tempera- 
ture (39.8°  F.  =  4°C.).  The  gram  is  divided  into 
centigrams  and  milligrams,  the  prefixes  centi  and  milli 
denoting  here,  as  always,  hundredths  and  thousandths 
of  the  unit.  A  mass  of  1,000  grams  is  called  a  kilo- 
gram. It  is  equivalent  to  2.2  English  pounds.  A 
new  five-cent  nickel  weighs  five  grams.  A  litre  of 
water  (1,000  cubic  centimetres)  weighs,  of  course,  one 
kilogram. 

The  unit  of  time  used  by  physicists  is  that  which  we 
use  in  daily  life,  the  second.  It  is  scnTRF  of  a  mean 
solar  day. 

Methods  of  measuring  these  fundamental  quantities 
are  explained  in  Part  II  of  this  book. 

The  system  of  units  based  upon  the  centimetre,  grain, 
and  second  is  known  as  the  c.  g.  s.  system. 

16.  Derived  Units.  —  The  unit  of  velocity  has 
received  no  special  name.  A  body  has  unit  velocity 
when  it  moves  unit  space  in  unit  time,  hence  in  the 
c.  g.  s.  system  the  unit  of  velocity  is  the  centimetre-per- 
second.  The  unit  of  momentum  is  the  gramr-centimetre- 
per-second.  The  unit  of  acceleration  is  the  centimetre- 
per-second-per-second.  The  unit  of  force  is  the  gram- 
centimetre-per-second-per-second.  Happily  it  has  received 


22  MATTER   AND   MOTION. 

a  shorter  name,  the  dyne.*  A  dyne  is  the  force  which, 
acting  for  one  second  upon  a  mass  of  one  gram,  will 
accelerate  its  velocity  one  centimetre  per  second.  For 
example,  a  body  is  acted  upon  for  one  second  by  the 
force  of  gravity.  It  is  required  to  find  the  intensity  of 
the  force  which  gravity  exerts  upon  it.  It  has  been 
found  that,  no  matter  what  the  mass  of  the  body  may 
be,  it  will  gain  in  one  second  a  velocity  of  980  cm. 
The  force  acting  upon  each  gram  of  the  body  is  there- 
fore 980  dynes. f  Forces  of  considerable  size  may  be 
conveniently  measured  in  grams  or  even  in  kilograms. 
The  value  may  then  be  expressed  in  dynes,  if  the  exact 
value  for  the  force  of  gravity  per  gram  of  mass  is 
known  at  the  place  where  the  measurement  is  made. 

The  exercises  which  follow  are  intended  to  illustrate 
and  fix  in  mind  the  principles  treated  in  this  chapter. 
For  convenience  the  various  formuhe  are  collected  and 
placed  at  the  head  of  the  exercises.  They  should  be 
fully  understood  and  thoroughly  memorized.  We  shall 
meet  them  again  and  again. 

Formulae. 

s  =  -         whence     (la)    I  =  st     (Ib)  t=- 

6  S 

a  =  —      whence     (2a)    v  —  at     (2b)  t  =  — 

M-=.  mv     whence      (3 a}  m  =  —    (3b^  v  =.  — 

v-      ^  m 

f  f 

f   =    ma    whence      (JicC\  m  =  -       (4^  a=.  — 

a  m 

*Dync  from  Greek  dyn,  forco. 

fThis  value  is  approximate.    It  varies  in  reality  from  977  to  984  at  different 
places  on  the  earth's  surface. 


EXERCISES.  23 

Exercises. 

1.  What  kinds  of  motion  are  illustrated  («)  in  the  sewing 
machine  ;  (&)  in  the  bicycle  ? 

2.  When  a  top  is  set  spinning  (rt)  what  keeps  it  spinning  ; 
(6)  what  makes  it  finally  stop  ?     What  law  of  motion  is  illus- 
trated ? 

3.  An.  ice  boat  has  sharp  runners,  so  that  it  cannot  slip  side- 
wise.     It  is   directed  toward  the  northeast.     If  the    wind   is 
blowing  ten  miles  an  hour  from  the  south  is  it  possible  for  the 
boat  to  be  driven  by  it  faster  that  ten  miles  an  hour  toward  the 
northeast  ?     Explain. 

4.  A  horse    pulls  a  loaded  wagon  exerting  a  force  of   300 
pounds,     (a)  How  man}'  dynes  is  that  ?     (6)  Where  is  the  300 
pounds  of  reaction  required  by  the  third  law  of  motion  ?     (c)  If 
he  were  on  smooth  ice  with  perfectly  smooth  shoes,  could  he 
do  it? 

5.  The  earth  rotates  once  in  twenty-four  hours.     Calling  its 
circumference  25,000  miles,  what  is  the  speed  of  a  point  on 
the  equator  (a)  in  miles  per  hour  ;  (&)  in  kilometres  per  hour  ; 
(c)  in  feet  per  second  ;  (d)  in  metres  per  second  ?     (e)  Why 
does  it  not  stop,  as  the  top  does  ? 

6.  (a)  Which  law  of  motion  is  illustrated  by  the  "kick"  of 
a  gun  ?     (6)  Why  does  not  the  gun  hurt  as  badly  as  the  bullet 
does? 

7.  A  body  starts  from  rest  and  gains  constantly  in  velocity 
until  at  the  end  of  ten  seconds  it  has  a  speed  of  20  metres  per 
second  :  (a)  What  was  its  acceleration  ?    (6)  How  far  did   it 
travel  ? 

Solution. 

(a)  by  equation  (2)  a  =  —  —  -—  =  2  metres  per  sec. 

0-1-20 
(6)  by  equation    (1)  I  =  st  =  -     -  X  10  =  100  metres. 


24  MATTER  AND   MOTION, 

NOTE.  —  The  speed  s  must  be  taken  as  the  average  speed, 
which  is,  of  course,  equal  to  half  the  sum  of  the  initial  and 
final  speeds  when  the  acceleration  is  constant. 

8.  A  ball  rolls  down  a  smooth  inclined  plane  10  metres  long 
and  6  metres  high  (see  Fig.  10).     We  may  resolve  the  weight 
of  the  ball  into  two  components  one   ||  and  one  _}_  the  plane, 
that  is,  one  in  the  direction  of  motion  and  one  at  right  angles 

to  it.  According  to  the  second  law 
of  motion  the  first  component  only 
produces  motion.  The  total  force  of 
gravity  upon  the  ball  would  produce 
an  acceleration  of  9.8  metres  per  sec- 
ond per  second.  What  acceleration 
will  the  component  ca  produce? 
NOTE.  —  Since  the  right  triangles  ABC  and  abc  have  the 

sides  AB 1  be  and  ab  1  BC  thev  are  similar,  and  —  = • 

ba       BA 

9.  A  body  falls  freely  under  the  influence  of  gravity.     It 
has   a   constant   acceleration   of    9.8   metres   per   second   per 
second  :  (a)  What  is  its  velocity  at  the  end  of  five  seconds  ? 
(6)  How  far  has  it  fallen  ? 

10.  A  man  who  can  lift  but  120  pounds  wishes  to  lift  a  200- 
pound  barrel  of  salt  into  a  wagon  3  feet  high.     If  he  is  to  roll 
it  up  a  plank,  how  long  a  plank  must  he  use  ? 

11.  A   car  weighing   1,000   kilograms    moves    east    with    a 
velocity  of  20  metres  per  second,  and  strikes  a  car  weighing 
600  kilograms  which  is  at  rest.     Both  cars   now   move   east : 
(a)    What   is   their  joint  momentum  ;  (6)  their  joint  velocity 
after  impact  ? 

12.  A  Spanish  ball  weighing  8  kilograms  and  moving  north 
at  the  rate  of  50  metres  a   second   meets   an   American  ball 
weighing  20  kilograms   and   moving   south  at  the  rate  of  20 
metres  per  second  :  What  is  the  velocity  of  the  combined  mass 
after  impact  ? 


CHAPTER  II. 
BALANCING  FORCES. 

17.  We  are  all  of  us  compelled  after  deliberation  to 
accept  the  truth  of  Newton's  first  law  of  motion,  and 
yet  our  own  observation  is  that  the  bodies  we  see  moving 
do  not  continue  in  motion  long  after  the  force  which 
sets  them  going  ceases  to  act.  The  condition  of  com- 
parative rest  rather  than  a  state  of  motion  seems  to  us 
the  normal  state  of  inanimate  objects.  To  see  a  moving 
body  come  to  rest  does  not  surprise  us.  It  is  when  a 
body  continues  long  in  motion  without  the  visible  appli- 
cation of  force  that  our  wonder  is  aroused. 

The  explanation  of  this  seeming  contradiction  is 
found  in  the  fact  that  moving  bodies  impart  motion  to 
all  bodies  which  they  touch  and  that  no  body  can  move 
at  the  surface  of  the  earth  without  touching  something. 
A  body  moving  in  the  air  must  push  the  air  to  one  side, 
it  must  put  the  air  in  motion.  It  will  lose  its  own 
motion,  therefore,  little  by  little  till  it  will  ultimately 
come  to  rest.  But  the  air  is  not  the  chief  hindrance  to 
motion  in  most  cases.  Gravity  is  constantly  acting. 
The  body  falls  to  the  earth.  If  it  rolls  along  uneven 
ground  it  will  be  lifted  out  of  the  hollows  by  its  own 
momentum  until,  in  rubbing  against  the  earth,  it  has  lost 
so  much  of  its  motion  that  it  finally  comes  to  rest  in  a 

25 


26  BALANCING   FORCES. 

hollow,  out  of  which  it  has  not  sufficient  momentum 
to  lift  itself  (see  Fig.  11).  This  rubbing  of  one  body 
against  another  is  known  as  friction.  It  is  in  reality  of 
the  same  nature  as  the  hindrance  due  to  the  large  ine- 
qualities of  the  surface  of  the  ground  just  mentioned. 
The  smoothest  surface,  when  examined  under  the  micro- 
scope, is  seen  to  be  covered  with  little  hollows  and 
projections.  When  one  body  rests  upon  another  the 
projections  of  the  one  fit  into  the  hollows  of  the  other 

(see  Fig.  12).    Before 

• — Q — y — • — *~p- ' — -£*—    the  first  body  can  slide 

FIG.  11.  along  the    surface  of 

the    second    it   must 
be    lifted    enough   to 


" 
J       let   the  projections 


FI0-12-  of  the  two   bodies 

clear  each  other.  While  the  motion  continues  there  is 
still  a  tendency  for  the  projections  to  drop  into  the 
depressions,  hence  friction  continues  until  the  moving 
body  comes  to  rest.  Polishing  the  surfaces  reduces  the 
height  of  the  projections  and  so  diminishes  friction.  The 
application  of  oil  or  graphite  fills  the  depressions  and  has  a 
like  effect.  The  heavier  the  moving  body  the  greater  the 
friction.  This  indicates  that  the  force  which  brings  the 
body  to  rest  is  gravity.  Now,  gravity  does  not  cease  to 
act  when  the  body  comes  to  rest,  as  you  may  convince 
yourself  by  holding  a  heavy  weight  in  your  hand  while 
the  hand  rests  upon  the  table.  In  fact,  the  condition 
of  rest  is  never  proof  that  no  force  is  acting  upon  the 


KINDS   OF  EQUILIBRIUM. 


27 


body  which  is  at  rest;  it  is  proof  that  the  forces  acting 
are  balanced,  that  is,  that  the  resultant  of  all  the  forces 
acting  upon  the  body  is  zero. 

This  balanced  condition  has  received  the  name 
equilibrium.  When  the  forces  acting  upon  a  body  are 
in  equilibrium  any  one  of  the  forces  is  equal  in  magni- 
tude and  opposite  in  direction  to  the  resultant  of  the 
remaining  forces. 

18.  Kinds  of  Equilibrium.  —  A  block  lying  upon  the 
ground  in  any  one  of  the  first  three  positions  shown 
in  Fig.  13  is  in  equilibrium  with  reference  to  gravity. 


FIG.  13. 


It  is  evidently  more  easily  overturned  in  1  than  in  2, 
and  less  easily  in  3  than  in  2  or  1.  It  might  be  bal- 
anced also  in  position  4,  but  in  that  position  it  would 
be  overturned  by  the  slightest  application  of  force  in 
almost  any  direction.  In  1,  2,  or  3  the  body  must  be 
lifted  to  overturn  it.  In  4  the  body  is  as  high  as  it 
can  be  and  touch  the  ground.  The  equilibrium  in  the 
first  three  cases  is  said  to  be  stable,  in  the  last  case 


28 


BALANCING   FORCES. 


unstable.  The  different  degrees  of  stability  in  the  first 
three  cases  may  be  more  clearly  shown  if  we  first  intro- 
duce the  idea  of  centre  of  mass.  Let  a  body  be  sup- 
ported by  a  string  attached  to  it  at  any  point,  and  let 
the  body  come  to  rest.  If  we  imagine  a  plane  drawn 
from  the  point  of  support  downward  toward  the 
centre  of  the  earth  the  body  will  be  divided  into  two 
parts,  such  that  the  force  of  gravity 
acting  upon  the  matter  in  one  part  of 
the  body  exactly  balances  the  force 
acting  upon  the  matter  in  the  other 
part  of  the  body  as  far  as  producing 
rotation  about  the  point  of  support, 
F  (Fig.  14),  is  concerned.  Let  a 
second  plane  of  the  same  sort  be 
drawn  through  F.  Now  support  the 
body  at  a  new  point,  F',  and  draw  a  vertical  plane 
through  that  point  also.  The  point  of  intersection  of 
the  three  planes  is  the  centre  of  mass  of  the  body. 

A  force  directed  through  the  centre  oj:  mass  has  no 
tendency  to  set  the  body  in  rotation.  A  force  applied 
to  the  body,  not  through  the  centre  of  mass,  does  tend 
to  produce  rotation.  When  a  body  is  supported  at  a 
point  directly  under  or  over  its  centre  of  mass,  gravity 
does  not  tend  to  make  it  rotate.  We  say,  therefore, 
that  gravity  acts  upon  a  body  exactly  the  same  as  it 
would  if  the  mass  of  the  body  were  collected  at  the 
centre  of  mass.  The  centre  of  mass  is  often  called  the 
centre  of  gravity.  When  a  body  as  a  whole  is  lowered 


FIG.  14. 


KINDS  OF  EQUILIBRIUM. 


29 


or  raised,  that  is  the  same  as  lowering  or  raising  its 
centre  of  mass.  In  Fig.  15  let  C,  Of  represent  in  each 
case  the  positions  of  the  centre  of  gravity  as  the  body  is 
overturned.  It  is  evident  that  the  distance  the  centre 


•-7C 


B~  -- 


3 


FIG.  15. 

of  mass  is  lifted  is  greatest  in  3,  which  is,  therefore,  the 
position  of  greatest  stability.  In  4  the  least  motion  will 
lower  the  centre  of  mass,  and  hence  the  equilibrium 
is  unstable. 

The  sphere  (1,  Fig.  16)  is  neither  stable  nor  un- 
stable, since  its  centre  of  mass  is  neither  raised  nor 
lowered  by  overturning  it.  The  equilibrium  is,  therefore, 
said  to  be  neutral. 


o 


w 

111 


FIG.  16. 

The    conditions   for   stability  are  that  the  centre  of 
mass   should   be  low   and   that   the    base,  that  is    the 


30  BALANCING   FORCES. 

figure  formed  by  connecting  the  points  of  support, 
should  be  large.  The  load  of  hay  (3,  Fig.  16)  is  less 
stable  than  the  load  of  stone  (2,  Fig.  16),  because  its  cen- 
tre of  mass  is  higher.  It  would  be  more  likely  to  upset, 
therefore,  on  a  rough  road,  than  would  the  load  of  stone. 

19.  Moments. — If  a  rigid  bar  be  supported  near 
its  centre  of  mass  it  will  be  in  neutral  equilibrium. 
Experiment  shows  that  two  equal  forces  acting  down- 
ward upon  the  two  sides  or  arms  of  the  lever  will  not 

produce  equilibrium, 

I  except  the  forces  be 

&/TL  applied  at  equal  dis- 

b      tances  from  the  cen- 

'  *vx^  /  R      tre  of  rotation,  and 

that  a  force  of  1  at  a 
distance  2  will  bal- 
ance a  force  of  2  at  a 

distance  1.  In  general  the  effect  of  a  force  in  produc- 
ing rotation  is  measured  by  the  product  of  the  magnitude 
of  the  force,  times  the  distance  of  the  line  of  application 
of  the  force  from  the  centre  of  rotation.  This  product 
is  called  the  moment  of  a  force. 

Moment  of  a  force  =  force  X  distance  from  centre. 
The  distance  to  the  line  of  application  of  the  force  is, 
of  course,  the  perpendicular  distance. 

The  condition  of  equilibrium  for  a  lever,  as  for  any 
body  free  to  rotate  about  a  fixed  axis,  is  that  the  sum  of 
the  moments  of  all  the  forces  acting  upon  it  shall  be  zero. 


THE  LEVER  BALANCE. 


31 


In  the  case  illustrated  in  Fig.  17  the  sum  of  the 
moments  of  the  forces  M  and  R  tending  to  rotate  the 
lever  toward  the  right  is:  4x3  +  3x6  =  30.  The 
moment  of  the  force  N  is  5  X  6  =  30  toward  the  left. 
Since  the  sum  of  these  moments  is  zero  the  lever  is  in 
equilibrium. 

20.  The  Lever  Balance. — The  force  of  gravity  at 
any  place  is  the  same  for  equal  masses.  A  lever  sus- 
pended at  a  point 
slightly  above  its 
centre  of  gravity  has 
attached  at  equal 


FIG.  19. 


It  will  again  be 
horizontal  when 
the  masses  i  n 
the  two  pans 
are  equal  (see 
Figs.  18,  19). 


FIG.  18. 

distances  from  its 
centre  pans  in  which 
the  masses  to  be 
compared  are  placed. 
The  lever,  or  beam 
as  it  is  called,  should 
be  horizontal  when 
the  pans  are  empty. 


<Qs~l 

6 


FIG.  20. 


32  BALANCING  FOECSS. 

When  large  masses  are  to  be  compared  with  smaller 
ones  a  lever  with  unequal  arms  (Fig.  20)  is  employed.    A 


FIG.  21. 

familiar  form  is  called  the  steelyards  (Fig.  21).  In  the 
large  scales  used  for  weighing  grain  or  coal  combina- 
tions of  several  levers  are  employed,  so  as  to  reduce 
the  size  of  the  standard  weights  used. 

21.  Universal  Gravitation.  —  With  truly  prophetic 
insight,  Sir  Isaac  Newton  conceived  that  the  force  of 
gravity,  which  acts  between  the  earth  and  all  bodies 
upon  its  surface,  reaches  also  to  the  moon  and  to  all  the 
planets  of  our  solar  system  —  indeed  to  the  limits  of  our 
visible  universe.  Certain  it  is  that  some  force  must  be 
constantly  acting  to  keep  the  moon  constantly  changing 
its  direction  as  it  revolves  about  the  earth  in  a  nearly 
circular  orbit.  Certain  it  is,  too,  that  gravitation  (of 
which  gravity  is  but  a  special  example),  acting  in  accord- 
ance with  the  laws  enunciated  by  Newton,  is  fully 
competent  to  keep  the  planets  in  equilibrium,  each  main- 
taining its  average  distance  from  the  sun  constant  from 
century  to  century. 


NEWTON'S  LAW  OF  GEAVITATION.  33 

22.  Newton's  Law  of  Gravitation.  —  Every  particle 
of  matter  in  the  universe  attracts  every  other  particle. 
^The  force  impelling  any  two  bodies  toward  each  other  is 
proportional  to  the  product  of  the  masses  of  the  two 
bodies  and  inversely  proportional  to  the  square  of  the 
distance  between  their  centres  of  mass.  ^ 

If  A,  B  are  two  spheres  whose  masses  are  mv  ra2,  and 
distance  between  centres,  r,  then  : 


where  Kis  a,  number  which  is  constant  for  any  set  of 
units  used  in  measuring  m  and  r.  It  is  the  attraction 
between  two  unit  masses  (1  g.  each)  whose  centres  of 
mass  are  unit  distance  (1  cm.)  apart.  In  dynes  JT— 
.000000065.  It  is  a  very  small  quantity,  as  one  can 
readily  see,  since  the  force  between  the  earth  and  a  body 
weighing  a  gram  is  only  980  dynes,  while  the  mass  of 
the  earth  is  so  great  as  it  is. 

The  force  between  two  kilogram  weights  at  a  distance 
of  ten  centimetres  is  so  small  that  very  delicate  instru- 
ments are  required  to  detect  it  at  all. 

The  force  between  the  earth  and  the  moon  is  much 
greater.  If  the  moon  could  be  stopped  in  its  revolu- 
tion about  the  earth  it  would  fall  at  once  to  the  earth. 
As  it  is,  the  force  of  gravitation  is  employed  in  deflect- 
ing the  moon's  motion  from  a  straight  line.  Gravita- 
tion, then,  tends  to  equilibrium.  In  the  case  of  bodies 
moving  in  orbits  a  sort  of  equilibrium  —  an  equilibrium 
of  motion,  in  which  the  average  distance  of  a  body  from 


34  BALANCING  FORCES. 

neighboring  bodies  remains  constant  —  has  been  already 
attained.  All  bodies  which  are  actually  falling  toward 
each  other  are  tending  to  a  condition  of  equilibrium. 
At  the  surface  of  the  earth  bodies  must  first  be  lifted 
before  they  can  fall.  When  a  body  is  once  lifted  we 
know  the  law,  we  learned  it  in  childhood :  "  All  that 
goes  up  must  come  down." 


SOME  PROPERTIES  OF  MATTER. 

23.  Elasticity.  — A  body  was  defined  in  Section  4  as 
a  portion  of  matter  having  a  definite  size  and  shape. 
Now  the  size  and  shape  of  a  body  may  be  changed 
somewhat  by  the  application  of  force  without  perma- 
nently altering  either  its  size  or  shape.  As  soon  as  the 
force  ceases  to  act  the  body  returns  to  its  original  con- 
dition. This  could  only  be  true  if  a  force  acted  to 
make  it  so  return.  We  call  the  forces,  concerned  in 
preserving  constant  the  size  and  shape  of  bodies  elastic 
forces,  and  the  property  which  different  substances 
possess  in  different  degrees  of  exhibiting  elastic  force 
we  call  elasticity. 

Any  force  which  acts  upon  a  body  so  as  to  change  its 
form  or  size  from  that  which  it  would  assume  under  the 
action  of  the  elastic  forces  is  called  a  stress.  The  effect 
of  a  stress  in  deforming  the  body  is  called  a  strain. 
When  the  stress  is  removed  elasticity  causes  the  body 
to  recover  from  the  strain  and  return  to  its  original 
form  and  size. 


POBES,  MOLECULES,  COHESION.        35 

If  a  metre  stick,  with  its  ends  resting  upon  two 
blocks  or  books,  have  a  kilogram  weight  placed  upon  it 
at  its  middle,  it  will  be  depressed  until  the  elastic  force 
of  the  wood  is  equal  to  one  kilogram,  when  it  will 
come  to  rest.  If  the  weight  be  now  removed  the  stick 
will  again  become  straight. 

The  air  in  a  boy's  pop-gun  is  compressed  against  an 
elastic  force  which  increases  until  the  wad  is  forced  out, 
when  the  air  returns  at  once  to  its  original  volume. 

24.  Pores,  Molecules,  Cohesion.  —  In  order  to 
understand  how  the  same  body  can  occupy  a  different 
amount  of  space  at  different  times  we  must  suppose 
that  before  the  air  in  the  pop-gun,  for  example,  was 
compressed,  there  were  portions  of  space  in  it  which 
were  not  occupied,  so  that  some  of  the  particles  could 
be  forced  into  these  pores.  There  are  excellent  reasons 
for  thinking  that  all  bodies  are  composed  of  very  small 
particles  which  have  been  called  molecules.*  The  pores 
are  the  spaces  between  the  molecules.  Two  forces  are 
constantly  acting,  one  to  draw  the  molecules  toward 
each  other,  one  to  push  them  apart.  The  force  which 
draws  the  molecules  together  is  cohesion  (called  ad- 
hesion when  it  acts  between  two  different  bodies). 
In  what  respects  cohesion  differs  from  gravitation 
is  not  yet  known.  It  is  effective  only  through  very 
small  distances.  Very  fine  iron  dust  will  remain  dust 

*  Mol/e-cule,  Latin  moleculum,  from  moles,  mass,  and  culum,  little  —  a  little 
mass. 


36  BALANCING  FORCES. 

though  its  particles  are  in  contact.  Under  heavy  pres- 
sure, however,  most  powdered  substances  will  cohere, 
especially  if  they  are  first  moistened  or  heated,  but 
the  structure  of  the  body  thus  formed  differs  from  the 
structure  of  the  substance  before  it  Avas  ground  to 
powder. 

The  force  which  balances  cohesion  is  called  heat.  It 
is  a  vibratory  motion  of  the  molecules  which  prevents 
their  coming  closer  together.  If  a  body  is  made  hotter, 
therefore,  its  molecules  are  driven  farther  apart  —  it 
becomes  larger.  If  the  body  loses  heat  it  grows  smaller, 
until  cohesion  and  heat  exactly  balance  each  other.  In 
a  solid  body  cohesion  so  far  predominates  over  heat  that 
the  molecules  retain  certain  definite  positions  \vith  ref- 
erence to  each  other.  If  enough  vibratory  motion  is 
imparted  to  the  molecules  in  the  form  of  heat,  cohesion 
will  be  so  far  overcome  that  it  Avill  no  longer  be  able  to 
keep  the  body  in  a  definite  shape,  the  substance  becomes 
liquid  and  under  the  influence  of  gravity  takes  the 
shape  of  the  vessel  in  which  it  is  contained,  its  upper 
surface  being  nearly  horizontal.  If  heat  be  still  further 
applied  the  liquid  will  expand  for  a  time,  but  if  a  suffi- 
cient amount  of  heat  is  imparted  to  the  body  the  par- 
ticles will  be  driven  so  far  apart  that  cohesion  loses  all 
control  and  the  body  becomes  a  gas. 

Most  substances  may  exist  in  any  one  of  these  three 
forms.  All  gases  are  now  liquefied  in  the  laboratory. 
All  solid  bodies  may  now  be  volatilized,  or  turned  to 
gas.  Some  substances  do  not  readily  take  the  liquid 


ELASTICITY  EXPLAINED. 


37 


FIG.  22. 


form  but  pass  at  once  from  solid  to  gaseous,  or  from 
gaseous  to  solid.  This  matter  will  be  discussed  more 
fully  under  the  subject  of  heat. 

In  Fig.  22  let  the  small  black  dots  at  a  represent  the 
position  of  the  molecules  of  a  small  body  at  a  given 
instant.  The  circles  may  represent  the  range  of  the 
motion  of  the  individual  molecules.  At  b  more  heat 
has  been  added  to  the  body,  the  molecules  require  more 
room  in  which  to  vibrate, 
the  body  is  larger.  At 
c  after  more  heat  has 
been  added  the  body  has 
become  a  liquid.  To 
represent  on  the  same  scale  the  gaseous  condition 
would  require  a  diagram  too  large  for  this  book. 

25.  Elasticity  Explained.  —  We  may  explain  in  the 
light  of  the  statements  just  made  the  phenomena  which 
we  have  ascribed  to  elastic  forces.  In  short,  heat  and 

cohesion  are  the  elastic  forces. 
In  Fig.  23  let  a  represent  a 
bar  of  steel.  Let  b  represent 
the  same  bar,  which  has  been 
bent  by  applied  forces  into  the 
form  shown.  The  molecules 
in  the  lower  row  are  nearer 
together  than  they  normally 
are,  those  in  the  upper  row  farther  apart.  Heat  is 
striving  to  drive  the  lower  row  apart>  cohesion  tends 


PIG.  23. 


38  BALANCING  FOKCE8. 

to  draw  the  upper  row  together.  In  other  words,  the 
body  tends  to  return  to  its  original  shape.  Elasticity 
of  shape  is  thus  seen  to  be  exactly  like  elasticity  of  size. 
In  both  cases  it  is  the  effect  of  forces  tending  to  prevent 
expansion  or  compression.  In  solids  different  parts 
of  a  body  may  be  compressed  or  expanded  unequally. 
In  liquids  and  gases  this  is  impossible,  hence  liquids 
and  gases  (both  of  which  are  called  fluids*)  have  no 
form  and  exhibit  only  elasticity  of  size  or  volume. 
Owing  to  the  freedom  of  motion  of  the  molecules  in 
fluids,  all  fluids  are  highly  elastic. 


26.  Liquids  in  Open  Vessels.  —  A  liquid  in  an  open 

vessel  is  acted  upon  not  only 
by  the  elastic  forces  but  by 
gravity.  Every  particle  of 
the  liquid  is  pressed  down- 
ward by  the  weight  of  the 
liquid  above  it  and  by  the 
weight  of  the  air.  Let  a 
(Fig.  24)  be  a  molecule  of 

water.  It  is  pressed  downward  by  the  weight  of  the  air 
and  the  water  above  it.  But  since  it  is  at  rest  it  must 
be  pressed  upward  by  an  equal  amount.  Moreover,  if 
it  were  not  pressed  from  all  sides  by  a  like  amount  it 
would  slip  out  of  its  place,  since  the  particles  of  a  liquid 
are  free  to  move  among  themselves.  The  fact  is  that 
the  molecules  <?,  d,  and  all  others  which  are  the  same 


Latin  fluere,  to  flow. 


LIQUIDS  IN  OPEN   VESSELS.  39 

distance  below  the  surface  are  pressed  downward  with 
the  same  force  that  a  is.  They  must  all  press  sidewise 
by  a  like  amount  or  the  liquid  would  not  be  at  rest. 
The  molecules  at  e,  d,  press  the  sides  of  the  vessel  with 
a  force  equal  to  that  exerted  upon  them  by  their 
neighbors  ,/,  g.  Any  surface  on  the  line  de  has  a 
pressure  upon  it  which  is  proportional  to  the  area  of  the 
surface,  but  is  the  same  for  all  equal  surfaces  for  that 
depth  and  is  alike  in  every  direction. 

Let  us  now  suppose  that  by  tilting  the  vessel  suddenly 
the  water  is  thrown  into  the  position  shown  in  Fig.  25. 
The  pressure  upon  c  is  less  than  that  upon  £,  conse- 
quently a  will  be  pushed 
away  from  b  with  a  greater 
force  than  it  is  pushed  away 
from  c.  The  same  is  true 
of  other  particles,  hence  they 
will  move  to  the  left  till 
equilibrium  is  restored,  that 
is,  until  all  parts  of  the 
liquid  surface  are  at  right  angles  to  the  direction  of 
gravity.  The  last  statement  presupposes  that  the  body 
of  liquid  is  so  large  that  gravity  is  the  principal  force 
acting. 

If  other  forces  are  acting  the  liquid  will  come  to  rest 
with  every  point  of  its  surface  perpendicular  to  the 
resultant  of  all  the  forces  acting  at  that  point. 

Let  us  first  consider  the  case  of  a  liquid  acted  upon 
by  gravity  only.  It  follows  from  the  principle  just 


FIG.  25. 


40 


BALANCING  FORCES. 


stated  that  in  any  number  of  communicating  vessels,  no 
matter  what  their  size  or  shape,  a  liquid  will  have  the 
same  level  (see  Fig.  26).  In  a  tea-pot  the  liquid  rises 


FIG.  26. 


as  high  in  the  spout  as  in  the  pot  itself  and  no  higher. 

The  pressure  of  any  particle  at  a  (Fig.  27)  due  to  the 

liquid  in  (7must  be  exactly  equal 
to  that  due  to  the  liquid  in  D,  or 
the  liquid  would  not  come  to 
rest.    If  the  height  of  the  liquid 
above  a  were  greater  in  D  than 
in  (7,  the  pressure  would  not  be 
equal    and  there  would  be  a 
FIG.  27. movement  from  D  to  0. 
We  may  now  sum  up  the  facts  in  regard  to  the  behavior 

of  liquids  in  open  vessels  in  the  following  statements : 

I.  The  pressure  at  any  point  in  a  liquid  is  the  same 
in  all  directions. 


CAPILLARY  PHENOMENA. 


41 


II.  The  pressure  at  any  point  in  a  liquid  of  uniform 
density,  due  to  the  weight  of  the  liquid,  is  proportional  to 
the  depth  of  the  point  below  the  surface  of  the  liquid. 

III.  The  surface  of  a  liquid  at  rest  is  perpendicular  at 
any  point  to  the  resultant  of  all  the 

forces  acting  at  that  point. 

If  we  now  examine  some  cases  in 
which  gravity  is  not  the  greatest  force 
acting  upon  the  liquid,  we  shall  see 
that  they  are  apparently,  but  not 
really,  exceptions  to  the  third  law. 


FIG.  28. 


27.  Capillary  Phenomena.  —  If  a 
piece  of    clean    glass  be   dipped    in 

water  some  of  the  water  will  adhere 
to  the  glass  when  it  is  removed  (Fig. 
28).  Here  adhesion  is  more  power- 
ful than  gravity,  and  the  surface  of 
the  water  may  be  almost  parallel  to 
the  direction  o  f 
gravity  (Fig.  28). 
Let  us  again  dip 
the  plate  of  glass  in  water  (Fig.  29). 
The  water  will  be  drawn  to  the  water 
on  the  glass  by  cohesion,  to  the 
glass  by  adhesion,  and  downward  by 
gravity.  Very  near  the  glass  adhe- 
sion is  most  powerful,  hence  the  re- 
sultant is  nearly  horizontal,  as  at  P  (Fig.  30) ;  a  little 


FIG.  29. 


FIG.  30. 


42 


BALANCING  FORCES. 


distance  from  the  glass  gravity  is  much  the  greatest 
force  acting,  while  at  P"  the  forces  are  nearly  equal  in 
magnitude. 

If  two  glass  plates  are  placed  very  near  each  other  in 
water  the  weight  of  the  liquid  between  them  is  small 
as  compared  with  the  adhesion  between  the  water  and 
the  glass,  hence  the  water  rises  to 
a  considerable  height  above  the 
level  of  the  rest  of  the  water  in 
the  glass.  In  a  very  fine  glass 
tube  the  weight  of  water  is  ex- 
ceedingly small  as  compared  to 
the  surface  of  glass  and  water  in 
contact.  In  such  tubes  water 
rises  to  a  height  of  several  centi- 
Such  tubes  are  called  capillary 


FIG.  31. 


metres    (a,  Fig.  31). 
tubes,*  and    the  phenomenon  of 
elevation  of  a  liquid  in  capillary 
tubes  is  called  capillarity. 

If  instead  of  clean  glass  we 
had  used  oily  glass,  to  which 
water  does  not  adhere,  or  if  in- 
stead of  water  we  had  used  mer- 
cury, cohesion  in  the  liquid  itself 
would  have  caused  a  depression 
of  the  surface  where  the  liquid  touched  the  glass,  as 
shown  in  Fig.  32. 


FIG.  32. 


*  Latin  capillus,  a  hair. 


SURFACE   TENSION.  43 

28.  Surface  Tension.  —  Water  falls  through  the  air 
in  drops.  Water  on  the  surface  of  the  leaves  of  plants 
to  which  it  does  not  adhere  collects  in  spheroidal 
masses.  Mercury  spilled  upon  the  table  collects  in 

spheroids,  the  smallest  of 
which    are    nearly    perfect 
FlG-  33-  spheres,  the  larger  being  flat- 

tened  by  their  greater  weight  (Fig.  33).  A  glass  rod 
or  a  stick  of  sealing  wax,  when  broken,  shows  sharp 
edges.  If  the  broken  end  of  the  glass  rod  or  the  stick  of 
wax  be  held  in  a  flame  to  soften  it  the  sharp  corners  are 
rounded  off.  How  can  these  phenomena  be  explained  ? 

The  liquids  in  question  (for  the  wax  and  glass 
are  melted  in  the  flame)  behave  as  if  their  surfaces 
were  stretched,  or,  in  scientific  language,  under  ten- 
sion. The  drop  of  water  has  least  surface  when  it 
is  a  perfect  sphere.  The  mercury  in  Tig.  32  is  pushed 
down  by  the  glass  plate,  just  as  if  there  were  a 
skin  of  rubber  over  the  surface.  For 
this  reason  all  capillary  phenomena, 
as  well  as  the  other  phenomena  just 
referred  to,  are  usually  explained  as 
results  of  surface  tensions.  It  remains 


to  be  explained  why  liquids  are  under  FIG-  &. 

tension  at  their  surfaces.  A  molecule  of  water  at  a 
(Fig.  34)  is  acted  upon  by  the  cohesion  of  all  of  the 
molecules  within  a  certain  small  distance  from  it, 
but  it  is  drawn  equally  in  all  directions  and  is,  there- 
fore, in  equilibrium.  A  molecule  near  the  surface, 


44 


BALANCING   FORCES. 


as  b  (Fig.  34),  is  acted  upon  only  from  one  side,  and  so 
there  is  always  a  force  acting  inward.  It  is  transmitted, 
of  course,  throughout  the  liquid  and  produces  the  same 
effect  as  would  a  thin  elastic  rubber  bag  stretched  over 
the  liquid.  We  know  that  if  we  push  in  a  stretched 
membrane  at  one  point,  it  tends  when  released  to  come 
back  to  a  flat  surface  again.  A  drop  of  water  or  of  any 
liquid  behaves  in  the  same  way.  The  surface  of  any 
liquid  will  tend  to  take  the  shape  that  will  make  it  the 
smallest  possible.  The  tension  is  along  the  surface, 
hence  when  the  surface  is  flat  there  is  no  tendency  to 


FIG.  35. 

motion.  If,  however,  the  surface  is  curved,  as  when  a 
tube  is  pushed  down  into  a  vessel  of  mercury  (c,  Fig. 
35),  the  tension  of  the  surface  tries  to  shorten  the  sur- 
face and  so  pulls  the  mercury  downward.  In  case  of 
a  wet  glass  tube  in  water  (a,  Fig.  35),  the  surface  is 
shortened  if  the  water  rises.  The  force  at  any  point  in 
the  surface  may  be  represented  by  a  line  tangent  to  the 
surface.  This  force  (if  the  surface  is  curved)  has  a 
component  acting  toward  the  centre  of  curvature,  the 


SURFACE    TENSION. 


45 


magnitude  of  which  is  greater  as  the  curvature  becomes 
greater.  This  fact  will  help  us  to  explain  why  a  soap 
bubble  which  has  been  flattened  by  fanning  it,  or  a 
drop  of  water  which  has  become  elongated  in  falling, 
tends  at  once  to  take  the  spherical  form,  that  is,  the 
form  in  which  the  curvature  is  the  same  at  every  point. 
This  is  also  the  form  having  the  least  surface  for  a 
given  volume. 

In  Fig.  36,  if  the  tension  at  B  is  resolved  into  two 
components,  one  toward  the  centre  and  one  at  right 
angles  to  it, 
we  see  that 
the  compo- 
nent BC  is 
greater  the 
greater  the 
curvature  of 
the  surface. 
In  the  case 
of  the  sphere 
the  curva- 
ture is  every- 
where equal. 
In  the  tubes 

(Fig.  35)  the  liquid  comes  to  rest  when  the  -difference 
in  pressure  due  to  a  difference  in  height  inside  and 
outside  the  tube  balances  the  tension  due  to  the  curvature. 

If  a  drop  of  water  be  drawn  into  a  wet  tube  which 
tapers  and  the  tube  be  then  held  in  a  horizontal  posi- 


FlG.  3G. 


46  BALANCING   FORCES. 

tion,  as  in  Fig.  37,  the  curvature  at  b  will  be  greater 
than  at  c  (being  equal  to  the  curvature  of  the  tube  at 
those  points),  hence  the  water  will  move  toward  b.  A 
drop  of  mercury  in  a  similar  tube  (see  Fig.  38)  will 
move  in  the  opposite  direction.  The  surface  at  b  (Fig. 
37)  must  be  thought  of  as  extending  along  the  inside 
of  the  tube  to  the  end  at  a.  When  the  liquid  has  been 

drawn  to  a  the 

.  37.  forces   will   be 

equal  at  every  point  in  the  surface.  The  mercury 
in  Fig.  38  will  move  till  the  tube  no  longer  presses  it 
out  of  the  spher- 
ical  shape  at  a. 
It  will  not  then 
be  perfectly  FIG  38- 

spherical,  because  of  its  own  weight,  but  the  curvature 
will  be  equal  on  its  two  sides  and  its  motion  will  cease. 

29.  Floating  Bodies.  —  If  a  body  when  immersed  in 
water  and  left  to  itself  rises  toward  the  surface,  there 
must  be  an  upward  force  acting  against  gravity  and  that 
force  must  be  greater  than  the  weight  of  the  body.  By 
the  second  law  stated  on  page  41  the  pressure  on  the 
bottom  of  the  body  is  greater  than  on  the  top,  since 
there  is  a  greater  height  of  water  above  b  (see  Fig.  39) 
than  above  a. 

If  the  body  is  a  cube  1  cm.  in  height  the  pressure  on 


ARCHIMEDES'   LAW.  47 

the  top  at  a  depth  of  6  cm.  is  6  grams,  but  the  pressure 
on  the  bottom  at  a  depth  of  7  cm.  is  7  grams.  The 
resultant  upward  pressure  is  therefore  1  gram,  which  is 
the  weight  of  the  water  displaced  by  the  body.  This 
fact  was  discovered  by  Archimedes,  who  gave  his  name 
to  the  law.  The  law  may  be  stated  as  follows : 

Archimedes'  Law.  —  A  solid  immersed  in  a  fluid  is 
buoyed  upward  by  a  force  equal  to  the  weight  of  the 
fluid  which  it  displaces. 

Since  the  volume  of  liquid  displaced  by  a  solid  is 
equal  to  the  volume  of  the  solid,  and  since  the 'volume 
of  a  gram  of  water  is  one  cubic  centi- 
metre, it  follows  that  the  loss  of  weight 
of  a  solid  when  immersed  in  a  liquid 
is  numerically  equal  to  the  volume  of 
the  solid.  If  the  liquid  weighs  the 
same  per  unit  volume  as  the  solid  does 

r1 

the  solid  will  come  to  rest  at  any  point 
in  the  liquid.  If  the  solid  is  lighter,  volume  for  volume, 
than  the  liquid  the  solid  will  rise  above  the  surface  of 
the  liquid  far  enough  so  that  the  portion  immersed 
displaces  an  amount  of  the  liquid  equal  in  weight  to 
the  weight  of  the  solid.  This  principle  is  made  use  of 
for  finding  the  volume  of  an  irregular  solid.  The  solid 
is  first  weighed  in  air,  then  weighed  again  immersed  in 
water.  The  difference  between  the  two  weights  in  grams 
is  equal  to  the  volume  in  cubic  centimetres  of  the  body. 

30.  Liquids  in  Closed  Vessels.  —  We  have  seen  that 


48  BALANCING   FORCES. 

the  pressure  at  any  point  in  a  liquid  due  to  the  weight 
of  the  liquid  is  transmitted  equally  in  every  direction. 
If  the  vessel  be  closed  by  a  movable  piston  so  that  ad- 
ditional pressure  may  be  applied  the  same  law  holds. 
Thus  the  pressure  applied  at  the  pumping  station  of  a 
city  water  system  is  transmitted  through  the  water  in 
the  pipes  to  all  parts  of  the  city,  forcing  the  water  to  the 
tops  of  high  buildings  or  at  lower  levels  delivering  it 
under  pressure  sufficient  to  run  a  water  motor.  The 
subject  will  be  further  discussed  under  machines. 

31.  Gases  in  Open  Vessels. — Gases  have  some 
properties  in  common  with  liquids.  They  transmit 
pressure  equally  in  all  directions.  Archimedes'  law  is 
true  of  liquids  and  gases  alike.  A  feather  weighing 
one  gram  is  as  heavy  as  a  gram  of  lead,  but  it  falls  to  the 
ground  much  more  slowly  because  it  displaces  a  large 
volume  of  air.  Liquids  and  gases  differ,  however,  in 
some  important  particulars.  Gases  have  no  definite 
surface,  hence  cannot  have  surface  tension.  A  gas 
seeks  to  diffuse  itself  indefinitely  because  its  particles 
are  too  far  apart  to  be  influenced  by  cohesion.  Let 
heat  be  removed  from  the  gas  so  that  its  particles  are 
not  driven  apart  when  they  chance  to  touch  each  other, 
or  let  pressure  be  applied  to  bring  the  particles  near 
together,  and  cohesion  takes  effect  at  once,  reducing  the 
gas  to  a  liquid. 

Liquids,  again,  have  a  very  definite  volume.  They 
are  almost  perfectly  incompressible.  Gases  are  very 


GASES  IN   OPEN    VESSELS. 


49 


easily  compressed,  the  volume  of  a  gas  being  determined 
by  the  pressure  which  it  sustains.  Every  open  vessel  at 
the  surface  of  the  earth  is  filled  with  air.  The  weight 
of  the  air  contained  in  any  given  vessel  varies  from  day 
to  day,  as  the  pressure  of  the  air  varies.  The  weight  of 
the  air  being  determined  by  the  pressure  of  the  air 
above  will  change  if  there  are  disturbances  in  the  air 
above  such  as  to  cause  the  air  to 
be  heaped  up  at  certain  places 
on  the  earth's  surface.  There 
are  such  disturbances,  known  as 
storms,  which  move  across  the 
country  from  west  to  east.  The 
pressure  of  the  air  often  varies 
as  much  as  two  or  three  per  cent 
}  in  two  days  during  the  passage 
of  a  storm.  A  diminution  of 
pressure  precedes  a  storm,  a  fact 
which  is  made  use  of  to  foretell 
the  approach  of  storms.  Let  us 
see  how  the  pressure  of  the  air 
may  be  measured.  Let  a  U  tube 
80  cm.  high  (Fig.  40)  be  filled  tfl 
about  half  full  with  mercury. 
The  mercury  will  come  to  rest 
FIG.  40.  with  its  two  surfaces  on  a  level.  FIG-  41- 
The  pressure  of  the  air  on  the  two  surfaces  is  equal. 
If  now  we  exhaust  the  air  from  the  left-hand  arm,  a,  of 
the  U  tube,  the  pressure  of  the  air  in  b  will  force  the 


50 


BALANCING   FORCES. 


mercury  up  in  a  until  the  weight  of  the  air  is  exactly 
balanced  by  the  column  of  mercury  ab'  (see  Fig.  41),  the 
portion  of  the  mercury  below  bbf  being  equal  in  both 
arms.  If  the  arm  a  be  now  closed  permanently  and  the 
arm  b  be  cut  off  to  a  convenient  length,  we  have  a 

means  of  meas- 
uring the  pres- 
sure of  the  air; 
for  as  the  pres- 
sure of  the  air 
increases  or  di- 
minishes  we 
have  but  to 
measure  the  dif- 
ference in  height 
of  the  two  mer- 
cury surfaces. 
Such  an  instru- 
ment is  called 
a  barometer.* 
The  air  may  be 
removed  from 
the  long  arm  by- 
Ming  the  entire 
tube  with  mer- 
cury, employing  a  rubber  tube  and  funnel  as  shown  in 
Fig.  42,  a,  and  then  inverting  the  tube.  The  same  end 
may  be  accomplished  by  using  a  straight  tube,  b  (Fig.  43), 

*  Greek,  bar  and  metron,  weight,  measure. 


FIG.  42. 


FIG.  43. 


PUMPS  AND  SIPHONS. 


51 


which  is  filled  and  then  inverted 
in  a  small  vessel  of  mercury,  c. 
The  mercury  will  fall  in  the  tube 
until  the  pressure  of  the  air  is 
exactly  balanced. 

It  will  balance  at  about  73 
cm.  to  78  cm.,  depending  on  the 
elevation  of  the  place  and  the 
condition  of  the  weather. 

32.  Pumps  and  Siphons. — 
(#)  If  water  were  used  in  a 
barometer  instead  of  mercury, 
it  would  rise  13.6  times  as  high 
as  mercury  does,  since  mercury 
is  13.6  times  as  heavy  as  water. 
The  common  pump  consists  of 
a  long  tube,  £,  the  lower  end  of 
which  dips  in  water,  while  the 
upper  end  is  provided  with  a 
close-fitting  piston  for  removing 
the  air.  A  device  called  a  valve, 
v,  allows  the  air  to  escape  when 
the  piston  is  pushed  downward, 
but  closes  when  the  piston  is 
lifted  so  that  no  air  enters.  .FIG*  **' 

The  air  which  remains  in  the  tube  is  under  diminished 
pressure,  and  water  will  rise  in  the  tube  till  the  pressure 
of  the  air  and  water  within  the  tube  balance  that  of  the 


52 


BALANCING   FORCES. 


air  on  the  outside.  A  second  valve,  v\  is  placed  in  the 
tube  below  the  piston,  so  that  another  portion. of  air 
may  be  removed,  and  so  on  until  the  water  comes  above 
the  piston,  where  it  flows  out  at  the  spout,  s.  Such  a 
pump  will  raise  water  about  30  feet. 

(6)  A  siphon  is  used  for  transferring  liquids  from 
higher  to  lower  levels  over  an  elevation.     It  consists  of 


FIG.  45. 

a  bent  tube,  which  is  first  filled  with  the  liquid  and  then 
inverted  with  one  arm  under  the  surface  of  the  liquid, 
the  arm  outside  of  the  vessel  being  lower  than  the  one 
within.  In  Fig.  45  the  pressure  at  0  in  the  direction 
of  CA  is  one  atmosphere  less  the  weight  of  the  column 
AC-,  the  pressure  at  C  in  the  direction  CB  is  one 
atmosphere  less  the  weight  of  the  column  BC.  The 


GASES  IN  CLOSED  VESSELS.          53 

pressure  is  greater  in  the  direction  CB  by  the  weight  of 
A'B,  and  the  liquid  will  flow  until  it  is  at  the  same  level 
in  both  vessels. 

33.  Archimedes'   Principle  applies  to  Gases.  —  A 

solid  immersed  in  a  gas,  as  all  bodies  near  the  surface 
of  the  earth  are  immersed  in  air,  is  buoyed  up  by  a 
force  equal  to  the  weight  of  the  air  displaced  by  it.  A 
pound  of  feathers  weighed  out  with  a  balance  and  iron 
weights  is  really  heavier  than  a  pound  of  lead.  Placed 
on  the  two  arms  of  the  balance  in  air  they  would  bal- 
ance each  other.  In  vacuum  the  arm  carrying  the  lead 
would  require  some  additional  weight  to  again  produce 
equilibrium. 

34.  Gases  in  Closed  Vessels.  —  A  gas  wholly  fills 
the  vessel  containing  it,  no  matter  how  large  the  vessel 
may  be.     The  pressure  of   a  given  body  of  gas  is  in- 
versely proportional  to  the  space  it  occupies,  or  what 
amounts  to  the  same  thing,  its  volume  times  its  pressure 
is  a  constant  quantity  : 

Volume  X  pressure  =  constant 
vp  —  k 


This  statement  is  known  as  Boyle's  Law.  It  does 
not  apply  to  vapors,  that  is,  gaseous  substances,  which 
by  a  small  increase  of  pressure  or  lowering  of  temper- 
ature would  be  liquefied,  for  when  a  vapor  begins  to 
liquefy  it  may  be  subjected  to  pressure  without  much 


54 


BALANCING  FORCES. 


diminishing  the  space  occupied  by  the  vapor.  Boyle's 
Law  is  said  to  apply,  therefore,  to  the  so-called  "  perfect 
gases"  at  ordinary  temperatures.  Air,  oxygen,  and 
hydrogen  are  good  examples  of  perfect  gases. 

Boyle's  Law  may  be  veri- 
fied experimentally  by  means 
of  a  bent  tube,  having  its 
shorter  end  closed  (see  Fig. 
46)  and  the  bend  of  the  tube 
filled  with  mercury.  When 
the  mercury  is  at  the  same 
level  in  both  arms  of  the  tube 
the  pressure  on  the  confined 
air  is  the  same  as  on  the  free 
air,  and  is  measured  by  read- 
ing the  barometer.  If  the 
open  arm  be  filled  to  any 
height,  as  A,  the  air  in  the 
closed  arm  will  be  compressed 
to  a  smaller  volume.  The 
height  of  the  barometer  plus 
the  height  of  A  above  B' 
measures  the  pressure  upon 
the  confined  air.  If  the  tube 
is  of  uniform  diameter  the  volume  of  the  air  is  propor- 
tional to  the  length  CB. 

35.  Air  Pumps. — If  part  of  the  gas  in  any  closed 
vessel  is    removed    the  remaining  gas  immediately  ex- 


FIG.  46. 


FLUIDS  I.V  CONTACT.     DIFFUSION. 


55 


pands  and  fills  the  entire  space.  Thus,  if  the  piston, 
p  (Fig.  47),  is  drawn  from  near  the  bottom  of  the  cylin- 
der, (7,  pushing  the  air  before  it,  the  air  in  the  receiver, 
R,  will  expand  and  occupy  the  cylinder.  If  R  is  twice 
as  large  as  C,  one  stroke  of  the  piston  removes  one 
third  of  the  air  in  R,  a  second  stroke  removes  one 
third  of  the  remainder,  and  so  on,  the  limit  of  exhaus- 
tion being  determined  by  the  degree  of  perfection  of 


the  joints  about  the  piston  and  valves.  Compression 
pumps,  like  the  common  bicycle  pump,  have  the  valves 
opening  inward. 

36.  Fluids  in  Contact.  Diffusion.  —  (a)  If  a  ves- 
sel containing  a  gas  be  opened  in  a  room  the  gas  will  at 
once  expand  till  it  is  equally  distributed  throughout 
the  room.  Tin's  fact  is  evident  when  the  gas  has  a 


56 


BALANCING  FORCES. 


marked  odor,  but  it  is  equally  true  in  any  case.  It  is 
also  true  that  the  air  in  the  room  will  enter  the  vessel, 
and,  in  time,  the  contents  of  the  vessel  and  the  room 
will  be  identical.  If,  however,  gases  differing  Avidely 
in  density  be  placed  in  the  same  vessel,  the  heavier  gas 
will  tend  to  go  to  the  bottom;  but,  unless  the  vessel  is 
very  large  indeed,  every  part  of  the  vessel  will  show 

the  presence  of  all  the  gases 
in  the  vessel.  The  percen- 
tage of  carbon  dioxide  present 
in  a  sleeping  room  is  always 
greater  near  the  floor  than 
near  the  ceiling.  This  pene- 
tration by  a  gas  of  a  space 
already  occupied  is  called 
diffusion. 

The  rate  of  diffusion  for 
heavy  gases  is  less  than  that 
for  light  ones.  If  a  porous 
battery  cup  (see  Fig.  48) 
be  fitted  with  a  large  cork 
through  which  passes  a  glass 
tube,  the  air  in  the  cup  will 
diffuse  outward  through  the  pores  of  the  clay  at  the 
same  rate  that  the  air  on  the  outside  diffuses  inward. 
Should  we  now  dip  the  lower  end  of  the  tube  in  a  glass 
of  colored  water,  the  water  will  rise  in  the  tube  to  the 
height  of  the  water  in  the  glass ;  but  if  we  place  over 
the  porous  cup  a  bell  glass  containing  a  light  gas  like 


FIG.  48. 


FLUIDS  IN  CONTACT.    DIFFUSION.  57 

hydrogen  or  illuminating  gas,  the  hydrogen  will  diffuse 
into  the  porous  cup  faster  than  the  air  diffuses  out,  the 
pressure  in  the  cup  will  be  increased,  and  bubbles  will 
be  forced  out  at  the  bottom  of  the  tube.  If  we  now 
remove  the  bell  glass  the  hydrogen  will  pass  out  through 
the  pores  of  the  cup  faster  than  the  air  passes  in,  the 
pressure  within  the  cup  will  be  diminished,  and  the 
water  will  rise  in  the  tube  higher  than  in  the  glass  and 
then  gradually  fall  when  equilibrium  has  been  restored 
between  the  gases  in  the  cup ;  that  is,  when  the  propor- 
tion of  hydrogen  and  air  is  the  same  inside  the  cup  as 
outside. 

(£>)  When  a  gas  is  in  contact  with  a  liquid  some  of 
the  gas  will  penetrate  the  liquid.  Thus  water  which 
has  been  exposed  to  air  contains  air  enough  to  keep 
fishes  alive.  Housekeepers  know,  too,  that  milk  left 
open  near  onions  absorbs  the  odor  emitted  by  the 
onions. 

(c)  Liquids  which  do  not  separate  of  their  own 
Accord  when  mixed  will  mix  by  diffusion  when  placed 
in  contact.  The  process  is  much  slower  than  in  the 
case  of  gases,  but  it  may  be  observed  without  difficulty 
by  introducing,  by  means  of  a  thistle  tube,  a  heavy  liquid, 
like  copper  sulphate  solution,  to  the  bottom  of  a  tall  jar 
containing  water.  If  the  tube  is  withdrawn  carefully, 
the  line  of  separation  beween  the  water  and  the  heavy 
copper  sulphate  solution  will  be  plainly  marked.  After 
the  jar  has  stood  quietly  for  a  day  or  two,  however,  the 
copper  sulphate  will  be  seen  to  have  diffused  into  the 


58  BALANCING  FORCES. 

water.  In  a  week  the  blue  color  will  have  reached  the 
very  top  of  the  water,  thus  making  evident  the  fact 
which  we  have  many  other  reasons  for  believing,  namely, 
that  the  particles  of  fluids  are  in  incessant  motion  among 
themselves,  and  are  not,  even  in  liquids,  confined  per- 
manently to  any  particular  part  of  the  body  of  fluid  of 
which  they  form  a  part. 

37.  Osmose.  —  If  two  solutions  are  separated  by 
parchment  paper  arid  one  wets  the  paper  while  the 
other  does  not,  the  former  will  diffuse  through  the 
paper.  Solutions  of  crystalline  substances,  like  sugar 
and  salt,  pass  readily  through  such  porous  paper,  while 
gums,  albumen,  and  the  like  do  not.  This  fact  is  often 
made  use  of  to  separate  substances  of  the  first  class 
which  are  in  solution  with  those  of  "the  second  class. 
Osmose,  as  this  property  of  diffusion  of  solutions 
through  porous  partitions  is  called,  is  of  especial  inter- 
est in  the  study  of  plant  life.  In  winter  the  fluids  of 
a  tree  are  allowed  to  diminish  in  amount  so  as  to  pre- 
vent injury  to  the  cells  by  freezing.  In  spring  the  roots 
take  up  moisture  from  the  soil,  passing  it  from  cell  to 
cell  by  the  process  of  osmose,  till  the  sap,  often  con- 
taining a  noticeable  amount  of  sugar,  reaches  the  tip  of 
the  farthermost  twig.  The  circulation  of  the  sap  of 
plants  is  thus  carried  on  without  that  complicated  sys- 
tem of  arteries  and  pumps  which  is  found  in  the 
higher  animal  organisms. 


EXERCISES.  59 

Exercises. 

13.  Why  do  steel  journals  run  more  easily  in  brass  than  in 
steel  bearings  ? 

14.  Mention  some  cases  where  friction  is  an  advantage. 

15.  (a)  Do  wood  and  other  vegetable  tissues  shrink  or  swell 
when  wet  ?     (fr)  Is  a  rope  longer  or  shorter  when  wet  ?     Ex- 
plain. 

16.  Is  a  three-legged  or  a  four-legged  stool  the  more  stable 
on  uneven  ground  ? 

17.  (a)  A  steelyard  has  the  hook  which  supports  the  weight 
one  half  inch  from  the  fulcrum.     How  far  apart  must  the  one 
fourth  pound  notches  be  placed  on  the  beam  if  the  bob  weighs 
one  half  pound  ?     (6)  Sketch  a  steelyard  which  will  weigh  to 
one  ounce. 

18.  The  force  of  gravity  on  one  gram  at  the  earth's  surface 
is  980  dynes.     What  is  the  force  on  the  same  mass  100  kilo- 
metres above  the  surface  of  the  earth  ? 


Suggestions  :  (5)  /=  — —£--=••  If  is  the  constant  of  grav- 
itation, and  ml7  m2  are  constant  in  value,  so  /  varies  inversely 
with  r2,  that  is,  /://::  -L  :  _L.  or  /  :  /' ::  r'2  :  r\  .-./  = 


In  this  example  r  —  6,400  km. 
r>  =  6,500  km. 

19.  Two  kegs  of  shot,  weighing  100  kilograms  each,  lie  on  the 
floor  with  their  centres  40  cm.  apart.     What  is  the  force  tend- 
ing to  make  them  roll  toward  each  other  ? 

20.  (a)  The  mass  of  the  moon  is  ^-  that  of  the  earth,  and 
its  diameter  0.273  that  of  the  earth.     What  is  the  value  of  / 
(the  force  on  one  gram)  at  the  surface  of  the  moon  ?     (&)  The 


60 


BALANCING  FORCES. 


mass  of  Mars  is  i|  that  of  the  earth,  and  its  diameter  0.534 
that  of  the  earth.     What  is  the  value  of  /  at  its  surface  ? 

21.  Why  does  a  hydrogen  balloon  float  in  air  ;  wood  in  water 
and  not  in  air  ;  iron  in  mercury  and  not  in  water  ;  an  iron  ship 
in  water  ?     Can  the  floating  of  clouds  in  air  be  explained  in  the 
same  way  as  the  foregoing  ? 

22.  Why  does  a  needle  float  if  carefully  placed  in  a  horizontal 
position  on  water  ? 

23.  Why  will  the  wood  of  a  ship  sunk  in  deep   water   no 
longer  float  if  set  free  ? 

24.  (a)  What  part  of  a  milldam  should  be  strongest  ?     (6) 
Need  it  be  stronger  if  the  pond  is  a  mile  long  than  if  it  is  but 
a  few  rods  long  ? 

25.  Explain  how  a  spoon  may  be  filled  heaping  full  of  water. 

26.  Why  does  not 
good  letter  paper 
make  good  blotting 
paper  ? 

27.  How  do  fishes 
rise  and  sink  in 
water  ? 

28.  The   capillary 
tube  in  Fig.  49  is  so 
small  that  the  water 

will  rise  in  it  to  a  height  of  10  cm.     If  it  were  bent  at  a  height 
of  8  cm.,  would  the  water  from  A  flow  over  into  B  ? 

29.  Why  does  boiled  water  taste  "flat"  even  after  being 
cooled  ? 

30.  A  withered  apple  placed  under  the  receiver  of  an  air 
pump  will,  when  the  air  has  been  exhausted,  swell  out  and 
look  plump  and  smooth.     Explain. 

31.  What  well-known  facts  are  best  explained  on  the  hypoth- 
esis that  matter  is  made  up  of  very  small  parts  (molecules) 
which  are  in  constant  motion  and  have  in  most  cases  some 
space  between  them  ? 


CHAPTER   III. 
HEAT. 

38.  The  Nature  of  Heat.  —We  have  seen  that 
many  phenomena  not  easy  to  explain  on  any  other  hy- 
pothesis are  explained  by  supposing  that  matter  consists 
of  molecules  which  are,  so  far  as  we  know,  always  in 
motion  among  themselves,  the  amount  of  motion  pres- 
ent in  the  molecules  of  any  body  being  the  cause  which 
determines  whether  the  body  is  at  any  instant  solid, 
liquid,  or  gaseous.  We  have  called  this  movement  of 
the  molecules  among  themselves  heat,  and  we  shall  now 
proceed  to  study  more  in  detail  the  phenomena  con- 
nected with  and  the  laws  which  govern  this  kind  of 
motion. 

It  is  to  be  observed  that  heat,  as  we  shall  use  the  term, 
is  a  motion  of  the  particles  of  matter  among  themselves 
and  is  therefore  confined  to  bodies,  and  may  be  trans- 
mitted from  one  part  of  a  body  to  another  and  from 
one  body  to  another  body  in  contact  with  it,  but  cannot 
be  transmitted  through  empty  space.  The  radiations 
which  come  to  us  from  the  sun  through  empty  space 
are  not  heat,  but  a  form  of  energy,  which  may  be  trans- 
formed into  heat  or  into  some  other  form  of  energy 
according  to  circumstances.  Radiant  energy  will  there- 
fore be  treated  in  another  place. 

61 


62  HEAT. 

39.  Effects  of  Heat.  —  O)  The  most  obvious  effect 
of  heat  is  the  sensation  of  warmth  produced  in  us  when 
a  hot  body  comes  in  contact  with  our  skin,  (ft)  A 
piece  of  iron  when  heat  is  being  continually  imparted 
to  it  may  become  too  hot  for  us  comfortably  or  even 
safely  to  touch,  and  finally  it  may  glow,  first  red,  then 
white.  Most  bodies  which  give  us  light  are  hot,  light 
being  a  form  of  motion  which  in  these  cases  results  from 
heat  and  accompanies  it. 

(c)  Heat  changes  the  size  of  bodies.     Except  in  spe- 
cial cases  (when  some  substances  pass  from  the  liquid 
to  the  solid  form,  for  example),  the  effect  of  heat  is  to 
cause  bodies  to  expand.    As  has  been  remarked  already, 
cohesion,  the  unknown  force  which  draws  the  molecules 
together,  is  opposed  by  heat.     If  the  amount  of  heat  is 
increased,  the  molecules  will  be  driven  farther  apart  — 
the  body  will  occupy  more  space. 

(d)  The  change  of  state  of  bodies  from  solid  to  liquid 
and  gaseous  has  already  been  referred  to. 

(e)  Many   chemical    changes    are    conditioned  upon 
heat.     The  very  combustion  which  is  the  source  of  heat 
and  light  in  a  candle   flame   cannot  occur   unless  the 
wick  of  the  candle  is  first  heated  much  hotter  than  the 
temperature  of  a  living   room.     When    a   gust    of  air 
blows  the  hot  gases  away  from  the  burning  wick  the 
flame  is  extinguished.     While  any  adequate  discussion 
of  chemical  changes  is  beyond  the  scope  of  this  book, 
we  may  well  pause  here  to  note  the  difference  betAveen 
physical  and  chemical  changes.     The  physicist  finds  it 


TEMPERA  T  UfiE.  6  3 

convenient  to  consider  matter  made  up  of  molecules, 
all  the  molecules  of  a  particular  substance  being  exactly 
alike  and  remaining  identical  throughout  all  the  various 
physical  changes  to  which  the  substance  may  be  sub- 
jected. The  chemist  goes  a  step  farther  and  imagines 
the  molecule  to  be  composed  of  atoms  which  may  be 
unlike,  though  all  the  atoms  of  any  one  of  the  funda- 
mental substances,  or  elements  as  they  are  called,  are 
identical.  When  a  molecule  is  separated  into  its  com- 
ponent atoms,  or  when  two  or  more  atoms  unite  to  form 
a  new  molecule,  a  chemical  change  has  occurred. 

The  body  to  which  heat  is  imparted  usually  becomes 
hotter,  or,  to  use  scientific  terms,  its  temperature  is 
raised. 

40.  Temperature.  —  The  term  temperature,  while 
not  easy  to  define,  is  used  in  a  very  definite  sense.  It 
is  used  to  express  the  relative  hotness  (or  coldness)  of 
bodies.  Any  two  bodies  are  at  the  same  temperature 
if,  when  they  are  in  contact,  neither  gains  heat  from  the 
other.  If  one  of  two  bodies  which  are  in  contact  im- 
parts heat  to  the  other,  the  one  which  imparts  heat  is 
always  at  a  higher  temperature  than  the  one  receiving 
it.  Amount  of  heat  must  not  be  confused  with  temper- 
ature. A  kettle  full  of  lukewarm  water  contains  more 
heat  than  a  cupful  of  boiling  hot  water,  though  the 
temperature  of  the  latter  is  the  higher.  We  are  accus- 
tomed to  form  judgment  of  the  temperature  of  bodies 
by  our  sensations.  We  call  bodies  hot,  warm,  cool, 


64  HEAT. 

cold,  etc.,  without  attaching  to  these  words  any  very 
exact  or  definite  meaning.  Our  sensations  of  heat 
depend  so  much  upon  the  condition  of  our  body  at  the 
time  of  making  the  observation  that  they  cannot  be 
relied  upon  to  furnish  exact  information.  Thus,  if  one 
hand  be  held  for  a  few  moments  in  ice  water  while  the 
other  is  held  in  water  as  hot  as  can  be  borne,  and  then 
both  hands  be  plunged  at  once  into  a  vessel  of  blood- 
warm  water,  the  latter  will  feel  warm  to  the  hand 
which  has  been  in  cold  water,  while  it  feels  cold  to  the 
hand  which  has  just  come  out  of  hot  Avater.  A  com- 
mon method  of  measuring  temperature  with  exactness 
depends  upon  the  expansion  of  bodies  by  heat. 

41.  Expansion.  Coefficient  of  Expansion.  —  All 
solids  expand  when  heated.  The  amount  of  increase 
in  volume  is  usually  too  small  to  be  detected  without 
the  use  of  some  special  device.  A  ball  which  passes 
easily  through  a  ring  when  both  are  cold  will  not  pass 
when  the  ball  has  been  heated.  If  while  the  ball  is  hot 
the  ring  be  heated  also,  the  ball  will  again  pass  through. 
Railway  rails  are  laid  with  space  between  the  ends  to 
allow  for  expansion  in  hot  weather. 

Liquids  expand  much  more  than  solids  for  the  same 
change  in  temperature,  while  gases  expand  still  more 
rapidly  than  liquids.  This  would  seem  to  be  explained 
by  the  weakening  of  cohesion  at  the  instant  when  the 
change  of  state  occurs  in  each  case.  Water,  from  some 
cause  not  yet  well  understood,  is  peculiar  in  that  it  first 


THERMOMETRY.  65 

contracts  on  being  heated  (starting  at  the  temperature 
of  ice)  and  then  begins  to  expand ;  after  passing  the 
temperature  of  maximum  density  it  expands  like  other 
liquids. 

The  perfect  gases,  like  air,  expand  at  a  uniform  rate. 
The  expansion  of  air  by  heating  plays  a  most  important 
part  in  the  heating  and  ventilation  of  houses  and  in  the 
circulation  of  the  atmosphere.  These  matters  will  be 
discussed  more  fully  later. 

The  expansion  per  unit  of  length  of  a  solid  for  one 
degree  change  of  temperature  is  called  the  coefficient  of 
linear  expansion.  The  rate  of  increase  in  volume  per 
unit  volume  for  one  degree  change  of  temperature  is 
called  the  coefficient  of  cubical  expansion.  It  is  three 
times  the  coefficient  of  linear  expansion.  Before  it  can 
be  determined  we  must  first  have  a  unit  of  temperature. 

42.  Thermometry.  —  Since  most  substances  expand 
uniformly  with  increase  of  temperature,  expansion  may 
be  used  as  a  measure  of  change  of  temperature.  Water 
freezes  at  a  certain  definite  temperature  and  melts  at 
the  same  temperature.  Water  boils  at  another  very 
definite  temperature  (for  a  given  atmospheric  pressure) <, 
These  temperatures  serve  admirably,  therefore,  as  points 
of  reference.  Two  scales  of  temperature  are  in  common 
use,  —  the  Fahrenheit,  'used  only  by  English-speaking 
people,  the  Centigrade,  used  universally  by  scientists. 
In  the  former  the  difference  of  temperature  between  the 
temperature  of  melting  ice  and  that  of  boiling  water  is 


66 


HEAT. 


divided  into  180  equal  parts,  called  degrees  Fahrenheit 
(180°  F.),  while  in  the  latter  the  same  difference  in 
temperature  is  divided  into  100  equal 
parts,  called  degrees  Centigrade  (100°  C.). 
Fahrenheit  took  as  his  zero  the  tempera- 
ture of  a  freezing  mixture  made  of  ice  and 
sal  ammoniac.  It  is  said  that  he  aimed 
thus  to  avoid  negative  readings,  as  this  was 
the  lowest  temperature  observed  by  him 
at  Dantzig  in  the  year  1709,  when  he  was 
making  his  experiments.  The  zero  Fah- 
renheit is  32°  below  the  freezing  point  of 
water,  which  makes  the  boiling  point  of 
water  180°  -f  32°  =  212°  F.  Celsius,  who 
devised  the  Centigrade  scale,  used  the 
freezing  point  of  water  as  his  zero,  thus 
making  the  boiling  point  of  water  100°  C. 
The  change  covered  by  a  Fahrenheit  degree 
is  therefore  ||{r ==  I  ^hat  coverec^  ty  a 
Centigrade  degree.  To  reduce  any  num- 
ber of  Fahrenheit  degrees  to  Centigrade 
degrees  we  must  therefore  multiply  by  |. 
The  reading  on  Fahrenheit  scale  is  32°  at 
the  freezing  point;  hence  to  reduce  Fah- 
renheit reading  to  Centigrade  subtract  32° 
and  multiply  by  |. 

(7)  Reading  C.  =  f  (Reading  F.  —  32°). 

(8)  Reading  F.  =  f  Reading  C.  +  32°. 
Fig.  50  makes  the  matter  plain. 


j 

C. 

L 

F. 

100^ 

=212 

_i 

L_2OO 

90_| 

Ll9O 

80-J 

L.I80 

Ll70 

704 

Ll6O 

Liso 

60J 

Ll40 

J 

Ll30 

50_! 

Ll20 

-f 

Ll  10 

40_:: 

LJOO 

30J 

L.9O 

Lao 

20  1 

L70 

i 

Leo 

IOJ 

Lso 

3 

L.40 

o_| 

L32 

-4 

L20 

lO_| 

Lio 

-; 

L,o 

J 

Lio 

30-j 

L20 

J| 

Lso 

4O    = 

LAO 

1 

FIG.  50. 


THERMOMETRY.  67 

The  thermometers  most  used  consist  of  a  glass  tube 
of  very  small  bore,  terminating  in  a  bulb  which  contains 
mercury  or  alcohol.  The  tube  was  sealed  at  a  temper- 
ature at  which  the  liquid  filled  both  bulb  and  tube.  At 
the  lowest  temperature  for  which  the  thermometer  is 
designed  the  liquid  still  extends  a  short  distance  into 
the  tube.  A  comparatively  small  expansion  of  the 
liquid  in  the  bulb  causes  a  movement  of  the  fine  thread 
in  the  tube  of  sufficient  amount  to  be  easily  seen.  The 
method  of  marking  a  thermometer,  or  calibrating  it  as 
it  is  called,  is  explained  in  Part  II. 

The  glass  which  contains  the  mercury  expands  only 
about  one  seventh  as  much  as  the  mercury.  Alcohol 
expands  more  than  five  times  as  much  as  mercury,  and 
therefore  makes  a  more  sensitive  thermometer  than  does 
mercury.  It  does  not  freeze  even  at  the  lowest  arctic 
temperatures,  while  mercury  solidifies  at  —  39°  C.  Al- 
cohol is  not  opaque,  however,  even  when  colored,  so  that 
an  alcohol  thermometer  is  more  difficult  to  read  than  a 
mercurial  thermometer. 

Air  expands  four  times  as  much  as  alcohol,  and  is, 
therefore,  very  suitable  for  measuring  small  changes  of 
temperature  over  a  small  range.  The  air  thermometer 
has  the  disadvantage  that  its  readings  vary  with  the 
atmospheric  pressure,  so  that  a  reading  of  the  barometer 
must  be  made  whenever  the  temperature  is  observed. 

The  difference  in  expansion  of  two  metals  is  some- 
times made  use  of  in  constructing  thermometers.  If 
a  strip  of  iron  and  a  strip  of  zinc  are  riveted  together 


HEAT. 


at  several  points  so  that  the  compound  bar  is  straight 
when  cold,  the  two  metals  will  expand  when  heated  and 
the  bar  will  be  curved  as  shown  in  Fig.  51.  If  the  end 
at  a  is  rigidly  fastened  to  a  support,  the  free  end  may 
be  made  to  transmit  its  movements  to  a  pointer,  which 
will  indicate  the  temperature  upon  a  dial.  The  dial 

may  be  calibrated  by 
comparison  with  a 
good  mercurial  ther- 
mometer. 

The  same  sort  of 
compound  bar  may 
be  made  to  close  alter- 
nately two  electric 
circuits  when  the  tem- 
perature of  the  room 
varies  any  given 
amount  above  or  be- 
low the  normal,  thus 
turning  off  or  on  the 
steam,  and  so  auto- 
matically regulating  the  temperature  of  the  room. 
Such  a  device  is  called  a  thermostat.  It  is  sometimes 
used  to  give  an  alarm  of  fire  if  the  temperature  reaches 
a  dangerous  point. 

43.  Law    of    Charles.     Absolute    Zero.  —  It    was 

proved  by  Charles  in  1787  that  at  constant  pressure  the 
volume  of  a  gas  increases  uniformly  with  its  temperature. 


FIG.  51. 


QUANTITY  OF  HEAT.  69 

The  rate  of  increase  is  the  same  for  all  gases  and  at 
0°  C.  this  rate  is  2*3,  or  0.00366. 

If  the  volume  is  kept  constant  the  pressure  increases 
at  this  same  rate.  A  quantity  of  air  which,  with  the 
barometer  at  760  mm.,  occupies  273  c.c.  at  0°  C. 
would,  if  the  pressure  were  kept  constant,  occupy  373 
c.c.  at  100°  C.  and  173  c.c.  at  -  - 100°.  If,  then,  we 
mark  an  air  thermometer  in  degrees  of  the  same  length 
as  for  the  Centigrade  scale,  but  call  the  freezing  point 
of  water  273°,  we  shall  have  a  scale  on  which  the  read- 
ings are  directly  proportional  to  the  volume  of  the  gas. 
Such  a  scale  is  called  an  absolute  scale.  Its  zero  is  at 
—  273°  C.,  a  temperature  which  of  course  can  never 
be  reached  in  the  laboratory.  It  is  believed  that  the 
spaces  between  the  stars  are  at  the  absolute  zero,  how- 
ever, since  there  is  no  evidence  for  the  presence  of 
matter  in  any  appreciable  amount  in  the  space  between 
us  and  the  sun,  except  within  a  few  hundred  miles  of 
the  earth  and  a  few  thousand  miles  of  the  sun. 

To  express  Centigrade  readings  in  absolute  we  have 
only  to  add  273  to  the  Centigrade  reading.  Thus 
50°  C.=  323°  absolute. 

44.  Quantity  of  Heat.  —  The  quantity  of  heat  re- 
quired to  raise  the  temperature  of  a  given  body  one 
degree  differs  with  the  kind  and  quantity  of  matter 
comprising  the  body.  The  unit  of  heat  is  the  heat 
required  to  raise  the  temperature  of  one  gram  of  water 
1°  C.  The  unit  of  heat  thus  denned  is  called  the 


70  HEAT. 

calorie.     It   requires  200  calories  to  raise  the  temper- 
ature  of  20  grams  of  water  10°. 

45.  Heat  Capacity.  Specific  Heat.  —  The  quantity 
of  heat  (measured  in  calories)  required  to  raise  the 
temperature  of  any  body  1°  C.  is  the  heat  capacity  of 
that  body.  The  quantity  of  heat  required  to  raise  the 
temperature  of  one  gram  of  any  substance  1°  C.  is  called 
the  specific  heat  of  that  substance.  It  is  a  quantity 
which  is  very  different  for  different  substances,  but  is 
constant  for  any  given  substance  through  a  moderate 
range  of  temperature.  The  specific  heat  of  iron,  for 
example,  is  0.113.  The  heat  capacity  of  an  iron  kilo- 
gram weight  would  be  1,000x0.113  =  113  calories, 
while  the  heat  capacity  of  one  litre  of  water  (weight 
one  kilogram)  is  1,000  calories,  or  nine  times  as  much. 
If  one  kilogram  of  iron  at  100°  C.  were  plunged  into 
one  kilogram  of  water  at  0°  C.,  the  temperature  of  the 
water  would  rise  10°  C.  and  that  of  the  iron  would  fall 
90°  C.,  since  the  two  substances  in  contact  tend  to  come 
to  equilibrium.  It  follows  that  by  putting  together 
known  masses  of  two  substances  at  different  (known) 
temperatures,  we  may  determine  the  relative  specific 
heats  of  the  substances ;  and,  if  the  specific  heat  of  one 
is  known,  the  specific  heat  of  the  other  may  be  found  at 
once.  For  the  quantity  of  heat  transferred  is  H=  mst, 
where  m  is  the  mass,  s  the  specific  heat,  and  t  the  change 
of  temperature.  But  the  heat  lost  by  one  body  exactly 
equals  the  heat  gained  by  the  other.*  That  is: 

*  Proper  precautions  must  be  taken  to  prevent  loss  to  the  air  or  other  bodies. 


FUSION.  71 

'=  Hf,  whence  mst  =  m's't'  and 


~  m't> 

This  method  of  determining  specific  heat  is  called  the 
method  of  mixtures. 

46.  Fusion.  —  If  we  heat  a  solid  like  ice  or  wax  it 
will  rise  in  temperature  for  a  time  and  then  begin  to 
melt,  the  temperature  remaining  constant  after  melting 
has  once  begun  until  the  whole  is  melted,  when  the 
temperature  again  begins  to  rise.  If  the  liquid  is  now 
allowed  to  cool  slowly  it  will  begin  to  solidify  at  the 
same  temperature  at  which  it  melted.  There  are  two 
important  differences  to  be  noted  in  the  solidification  of 
water  and  wax.  The  water  solidifies  in  crystals,  that 
is,  prism-like  bodies  of  definite  geometrical  form,  while 
the  wax  exhibits  no  structure  whatever. 

The  water,  probably  because  the  arrangement  of  the 
molecules  in  a  particular  geometrical  form  leaves  more 
space  between  them,  expands  in  solidifying,  while  the 
wax  contracts.  Those  metals,  like  iron  and  brass,  which 
expand  on  solidifying  make  excellent  castings  because 
they  fill  out  the  mold.  Gold  or  silver,  011  the  other 
hand,  must  be  minted  or  stamped  to  give  clear,  sharp 
outlines. 

The  fact  that  water  expands  on  freezing,  coupled 
with  the  fact  that  it  contracts  when  warmed  from  0°  to 
4Q  C.,  has  important  bearings  on  the  life  of  all  marine 
animals  in  particular,  and,  in  less  degree,  upon  the  life 


72  HEAT. 

of  all  animals  outside  the  Torrid  Zone.  If  ice  were 
heavier  than  water  the  first  ice  formed  on  a  lake  in 
autumn  would  sink  to  the  bottom,  another  layer  would 
form  and  sink,  and  the  process  would  continue  till  the 
lake  became  a  mass  of  ice.  In  spring  the  top  layer 
would  melt,  but  since  water  transmits  heat  downward 
very  slowly  the  lower  portions  would  probably  never 
thaw  out.  Let  us  consider  what  actually  happens. 
The  surface  water  in  autumn  cools  down  to  4°  C.,  its 
temperature  of  maximum  density,  and  sinks  to  the  bot- 
tom. Freezing  is  thus  delayed  till  the  entire  body  of 
water  is  at  4°,  when  the  portion  at  the  top  cools  to  0° 
and  freezes,  and  the  water  next  to  the  layer  of  ice  is 
slowly  cooled  to  the  freezing  point  and  added  to  the 
first  layer  formed,  the  entire  layer  of  ice  seldom  extend- 
ing more  than  a  few  feet  at  most  downward,  while  the 
great  body  of  water  never  freezes. 

47.  Vaporization.  —  Most  liquids,  if  left  exposed  to 
the  air,  gradually  disappear.  The  molecules  within  the 
liquid,  while  they  are  all  in  motion,  have  not  all  the 
same  velocity.  Some  of  those  which  are  moving  fastest 
make  their  way  through  the  surface  and  go  off  as  vapor ; 
the  liquid  slowly  evaporates.  Any  increase  in  temper- 
ature will  increase  the  number  of  particles  which  have 
velocity  enough  to  penetrate  the  surface  film  and  so 
increase  the  rate  of  evaporation.  It  is  obvious,  too, 
that  a  large  exposed  surface  is  favorable  to  rapid  evapo- 
ration. If  the  air  above  the  surface  is  at  rest  it  will 


EBULLITION.  73 

soon  become  saturated;  that  is,  it  will  be  so  full  of 
vapor  that  the  gaseous  particles  constantly  striking  the 
surface  of  the  liquid  will  make  their  way  into  the  liquid 
as  fast  as  those  within  escape.  When  the  space  above 
a  liquid  is  saturated,  evaporation  and  condensation  are 
going  on  at  the  same  rate,  which  amounts  to  the  same 
thing  as  if  evaporation  ceased.  Evaporation  is  hindered 
by  the  pressure  of  the  air  on  the  surface  of  the  liquid ; 
it  therefore  goes  on  very  rapidly  in  a  partial  vacuum,  a 
fact  which  is  sometimes  made  use  of  in  evaporating  the 
water  from  sugar  sap.  The  pans  are  covered  with  a 
tight-fitting  hood,  which  terminates  in  a  pipe  leading  to 
an  air  pump.  This  pump  removes  the  vapor  at  the 
same  time  it  diminishes  the  pressure,  thus  producing 
very  rapid  evaporation. 

Evaporation  goes  on  from  the  surface  of  solids  also, 
but  in  less  degree  than  from  liquids,  since  the  molecules 
have  less  freedom  of  motion.  A  block  of  ice  left  on 
the  shady  side  of  a  building  on  the  coldest  day  in 
winter  will  slowly  evaporate,  as  may  be  seen  from  the 
fact  that  its  corners  become  rounded.  A  lump  of 
camphor  left  exposed  to  the  air  will  slowly  evaporate, 
though  it  does  not  melt.  The  evaporation  of  a  solid  is 
called  sublimation.  % 

48.  Ebullition. — Evaporation  occurs  only  from  ex- 
posed surfaces.  If  a  liquid  be  heated  hotter  and 
hotter,  a  point  will  in  general  be  reached  where  vapor 
bubbles  will  form  on  the  surface  of  the  vessel  where 


74  HEAT. 

heat  is  applied  and  in  the  body  of  the  liquid  itselt. 
These  bubbles  will  at  first  form,  rise  into  the  cooler 
portions  of  the  liquid  and  collapse,  giving  forth  at 
times  a  note  like  the  "  singing  "  of  a  teakettle.  When 
the  whole  of  the  liquid  reaches  the  temperature  at 
which  the  bubbles  formed  they  will  rise  rapidly  to  the 
surface.  The  liquid  is  boiling,  and  if  heat  is  still 
applied  it  will  continue  to  boil  without  any  lise  in  tem- 
perature till  all  the  liquid  has  been  vaporized.  The 
boiling  point  of  a  liquid  is  definite  for  a  given  pressure 
of  the  atmosphere,  but  falls  with  diminished  pressure. 
On  the  top  of  Pike's  Peak  water  boils  at  187°  F.,  while 
at  sea  level  it  boils  at  212°  F.  It  is  not  worth  while, 
therefore,  to  attempt  to  cook  vegetables  by  boiling  in 
an  open  vessel  at  such  high  altitudes. 

In  a  locomotive  boiler  carrying  a  steam  pressure  of 
two  hundred  pounds  to  the  square  inch  water  boils  at 
382°  F.  This  fact  accounts  for  the  terrible  severity  of 
scalds  produced  by  escaping  steam  following  the  burst- 
ing of  a  boiler  in  a  railway  collision. 

49.  Distillation.  —  The  fact  that  different  substances 
boil  at  different  temperatures  makes  it  possible  to  sepa- 
rate two  liquids  by  heating  the  mixture  to  a  tempera- 
ture a  little  above  the  boiling  point  of  the  more  volatile 
liquid,  which  will  then  go  off  in  vapor.  The  vapor  is 
conveyed  in  pipes  through  cold  water,  where  it  con- 
denses. Alcohol  and  water  are  separated  in  this  way. 
Since  evaporation  occurs  at  temperatures  below  the 


EVAPOEATION  A   COOLING  PROCESS.  75 

boiling  point,  a  small  portion  of  Avater  vapor  goes  over 
with  the  alcohol.  The  process  may  be  repeated,  how- 
ever, and  the  portion  first  collected  from  the  second 
distillation  Avill  be  almost  perfectly  pure  alcohol.  A 
still  is  shown  in  Fig.  52. 

50.  Evaporation  a  Cooling  Process.     Latent  Heat. 

—  We  have  seen  that  when  a  liquid  is  evaporating  it  is 
the  molecules  having  the  highest  velocities  which  leave 


FIG.  52. 

the  liquid.  Since  the  temperature  of  the  liquid  de- 
pends upon  the  average  velocity  of  its  molecules,  it  is 
evident  that  the  liquid  must  be  cooled  by  evaporation. 
If  we  wet  the  bulb  of  a  thermometer  with  water  the 
temperature  of  the  bulb  will  fall  to  a  point  several 
degrees  below  the  temperature  of  the  room,  the  amount 
it  falls  depending  in  part  upon  the  amount  of  moisture 
in  the  air.  Indeed,  the  difference  in  reading  between 
the  wet  and  dry  thermometers  at  any  given  tempera- 
ture is  an  index  to  the  humidity  of  the  air.  When 


76  HEAT. 

water  is  boiling  the  heat  supplied  to  keep  it  boiling 
makes  good  the  loss  of  heat  by  vaporization.  Since 
this  heat  produces  no  sensible  rise  in  temperature,  it  has 
been  called  latent  (hidden)  Jieat.  When  the  vapor 
liquefies,  the  heat  which  kept  the  molecules  apart 
against  cohesion  becomes  again  sensible  heat.  The 
amount  of  heat  required  to  vaporize  one  gram  of  water 
at  100°  C.  is  536  calories,  or  5.36  times  as  much  as 
would  heat  the  same  quantity  of  water  from  0°  to  100°. 
Tins  quantity,  536  calories,  is  called  the  latent  heat  of 
vaporization  of  water  at  100°.  It  is  somewhat  more  at 
lower  temperatures  and  less  at  higher. 

A  similar  loss  of  sensible  heat  occurs  in  the  change 
of  state  of  water  from  solid  to  liquid,  though  much  less 
in  amount.  The  latent  heat  of  fusion  of  water  at  0°  C. 
is  80  calories.  This  is  equivalent  to  saying  that  the 
amount  of  heat  required  to  melt  a  quantity  of  ice  at  0° 
would  raise  the  temperature  of  the  same  quantity  of 
water  from  0°  to  8tf°. 

51.  Conduction.  —  If  one  end  of  an  iron  bar  is 
heated  in  the  fire  the  other  end  will,  after  a  time,  grow 
hot  also.  The  motion  imparted  to  the  particles  by  the 
hot  coals  is  soon  passed  along  to  adjoining  particles, 
which  in  turn  pass  it  to  those  farther  away.  This 
process  is  called  conduction.  Substances  differ  greatly 
in  their  power  of  conduction,  the  metals  being  good 
conductors,  while  wood,  wool,  cotton,  water,  and  air 
are  poor  conductors. 


HEAT  AND  HUMAN  LIFE. 


77 


52.  Convection.  —  Fluids,  like  air  and  water,  while 
they  are  poor  conductors,  possess  two  properties  which 
make  it  possible  for  them  to  convey  heat  with  consider- 
able rapidity :  (1)  they  have  freedom  of  motion  among 
their  particles,  and  (2)  they  expand  a  great  deal  when 
heated.  The  result  is  that  if  a  fluid  comes  in  contact 
with  a  heated  surface  which  is  below  it,  it  begins  to 
expand  as  soon  as  heated,  and,  being  lighter  than  the 
surrounding  fluid,  is  forced  upward  by  the  latter,  which 
takes  its  place,  thus  keeping  up  a  constant  circula- 
tion by  means  of 


which  heat  is  con- 
veyed to  great  dis- 
tances. The  pro- 
cess is  therefore 
called  convection. 
The  pointed  shape 
of  a  candle  flame , 
,is  due  to  the  con- 
vection currents 
formed  in  the  air 
(see  Fig.  53). 


\ 


FIG.  53. 


53.  Heat  t  id  Human  Life.  —  The  temperature  of 
the  human  body  does  not  vary  in  health  more  than  a 
•degree  or  two  from  98.6°  F.  Yet  we  live  in  a  climate 
where  variations  of  50°  occur  within  twenty-four  hourp 
and  the  extreme  variation  in  a  season  may  reach  150°. 
It  may  be  profitable  then,  by  way  of  review,  to  see  how 


78  HEAT. 

the  laws  of   heat  which  we  have  been    studying   beai 
upon  some  of  the  problems  which  face  us  every  day. 

54.  Weather.  —  Wind,  temperature,  rain,  clouds, 
change  from  day  to  day  and  furnish  a  never-failing 
topic  of  conversation.  Everybody  is  interested  in  the 
weather,  yet  most  of  us  do  not  observe  weather  changes 
as  carefully  or  understand  them  as  fully  as  we  might. 
Winds  may  be  simple  convection  currents  flowing 
toward  the  hottest  region  of  the  earth.  Such  are  the 
"  trade  winds  "  which  flow  toward  the  equatorial  regions 
and  the  sea  breezes  which  blow  toward  the  land  in  day 
time,  alternating  with  land  breezes  toward  the  sea  at 
night. 

The  winds  which  most  affect  the  weather  in  tern 
perate  regions,  particularly  away  from  the  sea,  are  the 
cyclonic  winds  which  blow  toward  an  area  of  low  baro- 
metric pressure.  Such  low  areas  move  across  the 
country  from  west  to  east,  followed  by  regions  of  high 
pressure,  about  once  in  five  or  six  days  on  the  average  ; 
and  it  is  by  following  the  movement  of  such  storm 
areas  that  the  officials  of  the  Weather  Bureau  are  able 
to  predict  in  advance  the  weather  changes  winch  are 
likely  to  occur  within  the  next  thirty-six  hours.  To 
the  east  of  a  low  area  the  winds  blow  from  the  east 
and  south  and  are  warm  and  laden  with  moisture.  As 
the  low  area  passes  we  encounter  west  and  northwest 
winds  and  the  temperature  falls.  The  cooler  north 
winds  condense  the  moisture  which  the  south  winds 


HEATING  AND    VENTILATION.  79 

bring,  hence  we  have  the  most  clouds  and  rain  when 
the  barometer  is  low,  folloAved  by  fair  and  cooler 
weather  as  the  barometer  rises.  Absorption  of  solar 
radiation  by  day  and  loss  by  radiation  at  night  will  be 
better  understood  when  we  have  studied  radiant  energy. 
We  protect  ourselves  against  weather  changes  in  two 
ways  :  we  wear  clothing  and  live  in  houses. 

55.  Clothing.  —  The  heat  of  our  bodies  is  maintained 
by  chemical  changes  which  occur  in  the  bodily  tissues. 
These  changes  are  in  the  nature  of  a  disintegration  or 
destruction  of   the  tissues,   the    loss   being   constantly 
made  good  from  the  food  which  we  eat.     The  supply 
of  heat  thus  furnished  is  very  irregular,  but  the  body 
is  provided  with  an  automatic  regulating  device  in  the 
pores  of  the  skin.     Moisture  is  constantly  exuding  from 
these  pores  and  evaporating,    thus   cooling   the   body. 
When  the  body  is  warm  the  pores  open  and  we  perspire 
freely;  when  we  are  cold  the    pores   close.     Clothing 
checks  the  circulation  of  air  near  the  skin,  and,  being 
made  of  non-conducting  materials,  retards  the  escape  of 
heat  from  the  body. 

56.  Heating  and  Ventilation. — When  a  house  is 
heated  with  a  hot-air  furnace  taking  the  air  from  out  of 
doors  the  rooms  do  not  lack  for  fresh  air,  but  the  heat- 
ing of  the  house  is  not  likely  to  be  as  satisfactory  as 
when  the  cold  air  is  supplied  to  the  furnace  from  the 
coldest  parts  of   the   house,  such   as   a   hallway  or  a 


80 


HEAT. 


northwest  room.  With  this  arrangement  the  cold  air 
flows  down  to  the  furnace  while  the  hot  air  from  the 
registers  finds- its  way  to  these  colder  parts  of  the  house 
to  supply  the  place  of  the  cold  air.  Fresh  air  may  be 
admitted  to  any  part  of  the  house  through  windows  or 


FIG.  54. 

through  openings  provided  for  the  purpose.     The  cir- 
culation is  shown  in  Fig.  54.. 

When  a  large  room  is  to  be  heated  by  a  stove  the  end 
may  be  often  much  more  satisfactorily  accomplished  if, 
instead  of  leaving  the  circulation  of  air  in  the  room  to 
chance,  with  the  chances  in  favor  of  stagnation  and  a 


HEATING  AND    VENTILATION.  81 

wide  variation  of  temperature,  the  stove  be  surrounded 
by  a  large  drum,  open  at  top  and  bottom  and  reach- 
ing from  near  the  floor  to  a  point  some  distance  above 
the  top  of  the  stove.  The  drum  may  be  in  two  parts 
hinged  together  for  convenience  in  getting  at  the  stove. 
A  little  reflection  will  show  that  the  drum  will  shield 
that  portion  of  the  room  nearest  to  the  stove  from  ex- 
cessive heat,  while,  by  setting  up  convection  currents,  it 
brings  the  cold  air  from  the  farthest  parts  of  the  room 
to  the  stove,  where  it  is  heated  and  returned  along  the 
ceiling,  thus  heating  the  room  uniformly  and  much  im- 
proving the  ventilation  at  the  same  time.  In  buildings 
heated  by  steam  or  hot  water  the  hot  fluid  circulates 
through  the  pipes  by  convection,  and  the  heat  is  dis- 
tributed by  placing  in  each  room  a  radiator  consisting 
of  a  coil  of  pipe  having  an  amount  of  surface  suited  to 
the  size  of  the  room.  In  the  matter  of  ventilation  these 
systems  are  inferior  to  the  hot-air  furnace.  Pipes  are 
sometimes  so  placed  in  the  ventilating  flues,  however, 
as  to  create  a  draught  and  ventilate  the  room. 

A  combination  of  both  systems  is  now  being  used  in 
cities  where  power  is  easily  procured  in  which  air  is 
driven  over  hot  steam  pipes  (in  summer  over  ice)  by 
means  of  a  fan  run  by  an  electric  motor  or  other  power 
and  forced  through  registers  to  all  parts  of  the  room. 
In  this  way  a  large  church  may  be  thoroughly  ventilated 
and  heated  in  half  an  hour  during  the  coldest  weather, 
or  made  comfortably  cool  with  a  few  hundred  pounds 
of  ice  in  summer. 


82  HEAT. 

Exercises. 

32.  It  is  desired  to  obtain  the  temperature  of  a  liquid.    What 
error  might  arise  (a)  if  the  thermometer  were  read  soon  aftei 
inserting  in  the  liquid  ;  (6)  if  the  thermometer  were  inserted 
only  far  enough  for  the  liquid  to  cover  the  bulb  ;  (c)  if  you  tool 
the  thermometer  out  of  the  liquid  to  read  it  ?     Explain. 

33.  Why  is  there  sometimes  frost  on  the  grass  in  autumr 
when  a  self -registering  minimum  thermometer  records  no  lowei 
than  36°  F.? 

34.  Explain  how  the  height  of  a  mountain  above  sea  leve' 
might  be  determined  with  a  thermometer.     Would  it  be  neces 
sary  to  make  observations  on  more  than  one  day  to  obtain  accu 
rate  results  ? 

35.  What  is  normal  blood   temperature    (a)  in   Centigrad* 
degrees  ;   (6)  in  absolute  degrees  Centigrade  ;  (c)  in  absolute 
degrees  Fahrenheit  ? 

36.  Why  is  extreme  heat  harder  to  bear  in  moist  climates 
than  the  same  temperature  in  dry  climates  ? 

37.  Why  is  extreme  cold  felt  more  in  moist  climates  than  ir 
dry? 

38.  Why  do  boiler  makers  use  red-hot  rivets  for  fastening 
the  plates  of  a  boiler  together  ? 

39.  Why  do  freezing  water  pipes  burst  ? 

40.  A  certain   metal,  if  thrown   cold   into  a  vessel   of  th< 
same    metal    molten,    floats.     Will    this    metal    make    sharj 
castings  ? 

41.  Why  do  we  say  that  when  smoke  falls  from  the  chimneys 
it  is  likely  to  rain  ? 

42.  (a)  Why  does  a  pitcher  containing  cold  water  "sweat' 
in  summer  ?    (6)  A  kerosene  lamp  set  away  clean  is  often  founc 
covered  with  a  film  of  kerosene.     Is  this  fact  analogous  to  tht 
"  sweating  "  of  the  pitcher  of  water  ? 


EXEECI8E8.  83 


43'  An  air  thermometer  (see  Fig.  55)  was  filled  with 
water  to  the  point  A.  When  dipped  in  a  beaker  of 
hot  water  it  first  fell  to  B  and  then  rose  to  C ;  taken 
out  and  plunged  in  cold  water  it  rose  to  D  and  then 
fell.  Explain. 

44.  In  hot  weather  metal  objects  feel  hotter  than 
wooden  objects  which  are  beside  them  in  the  sun. 


Are  they  really  very  much  hotter,  and  if  not,  why  do 
they  seem  so  ?     What  is  the  case  in  cold  weather  ? 


45.  Ice  houses  are  made  with  double  walls  having 
sawdust   between  them.     Would  a  wall   of  logs   the 

same  thickness  as  the  double  rilled  wall  serve  as  well  ?  . 

46.  How  would  it  be  possible  at  high  altitudes  to 


cook  vegetables  by  boiling  ? 

47.  Tubs  of  water  are  often  placed  in  a  cellar  to 
prevent  vegetables  from  freezing.     Explain. 

48.  Why  are  the  extremes  of  heat  and  cold  greater 
in  the  middle  of  a  continent  than  on  the  coast  ? 

49.  Why  do  clothes  dry  faster  on  a  windy  day  than 
when  the  air  is  calm  ? 

50.  A  litre  of  air  at  746  mm.  pressure  and  20°  C. 
has  what  volume  under  standard  conditions  ;  namely, 
760  mm.  and  0°  C.? 

51.  An  iron  rail  is  30  feet  long  at  20°  F.     What  is 
its  length  at  120°  F.?     Iron  expands  0.0000064  of  its 
length  for  each  degree. 


B 


CHAPTER   IV. 
ELECTRICITY  AND   MAGNETISM. 

57.  Introductory  Remarks. — While  the  laws  gov- 
erning the  group  of  related  phenomena  which  we  desig- 
nate as  electrical  and  magnetic  are  very  well  known,  it 
must  be  borne  in  mind  that  we  are  as  yet  not  able  to 
give  any  very  satisfactory  answer  to  the  question :  What 
is  electricity?  To  be  sure,  we  have  not  the  slightest 
notion  what  cohesion  is  or  what  gravitation  is.  The 
words  cohesion  and  gravitation  are  but  symbols  for 
what  we  choose  to  designate  the  causes  of  two  classes 
of  phenomena.  Our  inability  to  define  more  exactly 
these  forces  need  not  stand  in  the  way  of  our  study  of 
the  phenomena  connected  with  them.  Indeed,  it  is 
only  by  frankly  admitting  our  ignorance  and  seeking  to 
find  out  as  large  a  body  of  facts  as  possible  that  we 
may  ever  hope  to  reach  a  knowledge  of  the  ultimate 
nature  of  such  fundamental  entities  as  matter,  electric- 
ity, and  gravitation. 

The  average  student  has  not  so  large  a  fund  of 
observation  and  experience  from  which  to  illustrate 
electrical  phenomena  as  he  had  in  the  subjects  treated 
in  preceding  chapters.  He  will  find  it  necessary,  there- 
fore,  to  observe  closely  and  remember  carefully  every 
new  fact  which  comes  in  his  way. 

84 


MAGNETIC  POLES.  85 

There  is  every  reason  why  all  intelligent  people 
should  wish  to  understand  that  wonderful  agent  which 
is  to  play  so  important  a  part  in  the  life  of  the  century 
now  opening,  if  indeed,  as  we  must  suppose,  the  dis- 
coveries of  the  last  fifty  years  have  but  opened  the  way 
to  still  more  wonderful  discoveries  yet  to  be  made. 

58.  Magnets.  —  It  was  long  ago  discovered  that  cer- 
tain specimens  of  iron  ore  had  the  power  to  attract  bits 
of  iron  to  themselves.     From  the  country  where  these 
"  lodestones  "  *  were  most  frequently  found  —  Magne- 
sia—  they   were    called    magnets.     It   was    very    early 
known,  too,  that  if  a  bar  of  steel  be  rubbed  with  a  lode- 
stone  or  natural  magnet  it  will  itself  become  a  magnet, 
and  that  such  artificial  magnets  will,  when  suspended, 
point  nearly  north  and  south.     An  elongated  mass  of 
lodestone  will  do  the  same  thing,  but  the  fact  is  much 
more  readily  observed  with  a  long  and  slender  bar  like 
a  compass    needle.     The  mariner's  compass  is  such   a 
bar  or  needle  supported  on  a  pivot  so  as  to  move  freely 
in  a  horizontal  plane.     It  was  in  use  before  the  time  of 
Columbus. 

59.  Magnetic  Poles.  —  If  we  dip  a  slender  bar  mag- 
net, like  a  magnetized  knitting  needle,  into  iron  filings, 
the  filings  will   adhere  to   the    magnet    only  near   the 
ends,  AD  and  OB  (Fig.  56).     If  we  cut  off  that  por- 
tion at  each  end  to  which  the  filings  adhered  and  dip 
the  magnet  again  in  filings,  the  filings  will  again  adhere 

*  Lode,  a  vein  of  iron. 


86  ELECTRICITY  AND  MAGNETISM. 

to  the  ends  of  the  portion  DC  which  remains.  It 
would  appear,  therefore,  that  while  the  magnetism 
manifests  itself  only  at  the  ends  of  the  bar,  it  is  really 
present  in  other  parts  of  the  bar,  or  it  would  have 
disappeared  when  the  ends  were  cut  off.  Had  we  cut 

B 


D 


FIG.  56. 

the  bar  at  its  middle  point  we  should  have  obtained 
two  magnets,  each  attracting  filings  at  its  ends.  More- 
over, the  two  pieces,  AD,  CB,  cut  from  AB  behave 
exactly  like  the  original  magnet.  The  ends  of  a  bar 
magnet  are  called  poles.  If  we  suspend  a  bar  magnet, 
and,  after  allowing  it  to  come  to  rest,  mark  the  end 
which  points  to  the  north,  we  shall  find  that  end  which 
we  have  marked  will  always  be  the  north-seeking  end 

N  _  S  N  _  S   N  _  S 
FIG.  67. 

of  the  magnet  ;  that  is  to  say,  the  two  poles  of  the  mag- 
net are  unlike.  If  A  was  a  north-seeking  pole  in  the 
magnet  shown  in  Fig.  56,  the  arrangement  of  poles  in 
the  fragments  would  be  as  shown  in  Fig.  57, 

60.  Attraction  and   Repulsion.  —  When  the  N-end 
of  a  magnet  is  brought  near  the  N-end  of  a  suspended 


INDUCED  MAGNETISM.  87 

magnet,  like  a  compass  needle,  the  N-end  of  the  needle 
is  repelled  and  the  S-end  attracted.  In  like  manner,  a 
S-pole  repels  a  S-pole  and  attracts  a  N-pole.  Briefly 
stated,  like  poles  repel,  unlike  attract. 

61.  Law  of  Force.  —  We  may  define  unit  pole  as 
follows  :  let  two  poles  be  of  equal  strength,  such  that 
at  a  distance  apart  of  one  cm/  they  repel  each  other 
with  a  force  of  one  dyne,  they  are  unit  poles.  Coulomb 
proved  that  the  law  of  force  is  similar  to  the  law  of 
gravitation:  two  mag- 
netic poles  of  strength 
m  units  and  mf  units 
respectively  at  a  dis- 
tance r  cm.  apart  re- 
pel or  attract  each 
other  with  a  force 
proportional  to  the 
product  of  their  FIG.  58. 

strengths  divided  by  the  square  of  the  distance  apart. 


62.  Induced  Magnetism.  —  Magnets  are  commonly 
made  by  rubbing  steel  bars  upon  a  magnet.  A  piece 
of  soft  iron  needs  only  to  be  brought  near  a  magnet  to 
become  itself  a  magnet  for  the  time  being,  j  Thus  the 
nail  in  Fig.  58  attracts  the  second  nail  and  holds  it  so 
long  as  the  former  is  in  contact  with  the  magnet.  If 
we  remove  the  magnet  the  nails  no  longer  cling 
together.  Hard  steel  is  not  so  easily  magnetized  as 


88 


ELECTRICITY  AND  MAGNETISM. 


soft  iron,  but  the  steel  retains  its  magnetism  after 
removal  from  the  magnet,  while  the  iron  does  not,) 
This  fact  is  analogous  to  the  fact  that  it  is  less  easy  to 
put  an  edge  or  a  point  on  hard  steel  than  on  soft  iron, 
but  the  edge  or  point  will  be  retained  correspondingly 
better  by  the  steel.  Magnetic  attraction  occurs  only 
between  magnets  in  every  case,  for  the  pieces  of  soft 


FIG.  59. 

iron  all  become  magnets  by  induction,  as  it  is  called, 
and  are  then  attracted.  The  induced  pole  nearest  the 
magnet  is  always  of  opposite  kind  to  that  which  in- 
duces it.  It  is  evident,  therefore,  that  a  magnet  cannot 
repel  soft  iron,  since  unlike  poles  always  attract. 

63.  The  Magnetic  Field.  —  A  magnet  moves  a  mag- 
netic needle  without  touching  it.     The  needle  may  be 


THE  MAGNETIC  FIELD.  89 

enclosed  in  a  glass  bottle.  The  magnet  still  acts  upon 
it.  The  air  may  be  exhausted  from  the  bottle.  The 
magnet  acts  exactly  as  before.  This  space  surrounding 
a  magnet,  witliin  which  a  magnet  cannot  come  without 
being  influenced  to  take  a  particular  direction  and 
within  which  also  every  bit  of  soft  iron  becomes  tempo- 
rarily a  magnet,  is  called  the  magnetic  field.  It  extends 


FIG.  60. 

in  all  directions  from  the  magnet,  growing  weaker  as  we 
go  farther  from  the  magnet,  in  accordance  with  the  law 
of  force  stated  in  Section  61. 

If  we  place  a  piece  of  glass  or  stiff  cardboard  on  a 
magnet,  dust  iron  filings  over  the  glass  and  gently  tap 
it,  the  iron  filings  will  arrange  themselves  end  to  end, 
the  direction  of  each  bit  of  iron  being  the  direction  of 
the  resultant  of  the  two  forces  from  the  two  poles  of 


90 


ELECTRICITY  AND  MAGNETISM. 


the  magnet.  Fig.  59  is  taken  from  a  photograph  of 
such  a  chart  of  the  magnetic  field  made  by  using  a 
photographic  plate  instead  of  the  piece  of  glass.*  Fig. 
60  shows  the  field  between  unlike  poles,  while  Fig.  61 
shows  the  field  between  like  poles. 

64.  Magnetic     Induction     Explained.  —  We    have 
seen  that  a  magnetized  knitting  needle  may  be  divided 


FIG.  61. 

into  a  great  many  short  parts,  and  each  little  piece  will 
be  a  magnet.  This  suggests  that  every  molecule  of  a 
magnet  is  itself  a  magnet  —  nay  more,  that  every  mole- 
cule of  iron  is  a  magnet.  In  a  bar  of  soft  iron  the  mol- 
ecules, being  magnets  and  more  free  to  turn  than  in  the 

*The  work  was  of  course  performed  in  a  dark  room.  When  the  filings  were 
in  position  an  electric  light  was  turned  on  for  a  second.  The  filings  were  then 
brushed  off,  and  the  plate  developed. 


MAGNETIC  SUBSTANCES.  91 

steel,  take  the  most  stable  position  possible  ;  that  is, 
every  N-pole  will  get  as  near  a  S-pole  as  possible.  In 
this  position  the  force  of  any  N-pole  is  neutralized  by 
the  S-pole  near  it  as  far  as  producing  any  effect  outside 
the  iron  is  concerned.  If  a  strong  magnet  is  brought 
near  the  iron  it  turns  the  little  molecular  magnets  all 
one  way,  and  the  molecules  at  the  ends  will  attract, 
while  those  at  intermediate  points  in  the  bar  still  neu- 
tralize each  other.  Fig.  62,  a,  shows  the  arrangement 


b 

FIG.  62. 

of  the  molecules  before  the  magnet  is  brought  near ; 
£>,  while  the  magnet  is  in  the  neighborhood. 

When  a  steel  knitting  needle  has  been  given  a  single 
stroke  with  a  magnet  it  is  a  weak  magnet,  since  only 
part  of  the  molecules  have  been  arranged.  By  further 
stroking  all  may  be  made  to  point  in  the  same  direction, 
after  which  no  magnet,  however  strong,  could  induce 
any  more  magnetism  in  the  needle  than  it  already 
possesses. 

65.  Magnetic  Substances.  —  We  have  hitherto 
spoken  only  of  iron  as  a  substance  capable  of  magnet- 


92  ELECTRICITY  AND  MAGNETISM. 

ization.  It  is  not,  however,  the  only  magnetic  sub- 
stance. Nickel  and  cobalt  are  magnetic  to  a  small 
degree  as  compared  to  iron,  but  to  a  large  degree  as 
compared  to  air,  water,  and  most  other  substances. 
Most  substances  are  so  slightly  magnetic  that  their 
magnetic  properties  can  be  detected  only  by  means  of 
very  delicate  instruments. 

66.  The  Nature  of  the  Magnetic  Field.  —  While  air 
or  other  substances  which  may  happen  to  lie  within  the 
field  of  force  of  a  magnet  are  influenced  by  that  field, 
they  are  in  no  way  necessary  to  its  existence.  The 
force  which  is  transmitted  from  a  magnet  to  a  compass 
needle  is  not  conveyed  by  means  of  the  air.  We  can- 
not think  of  a  body  being  set  in  motion  without  con- 
tact, direct  or  indirect,  with  the  body  which  sets  it  in 
motion.  It  is  believed  that  all  motions  which  are  trans- 
mitted from  one  body  to  another  not  in  contact  with 
it  are  conveyed  through  a  medium  which  pervades  all 
space,  even  the  space  occupied  by  matter.  This  medium 
which  is  very  elastic  is  called  the  ether.  The  ether  in  a 
magnetic  field  is  supposed  to  be  in  a  state  of  strain  due 
to  a  stress,  which  the  magnet,  in  some  way,  puts  upon 
it.  We  are  familiar  with  the  transference  of  stress  in 
an  elastic  fluid  from  one  body  to  another  in  the  fluid. 
The  bullet  in  an  air  gun  has  motion  imparted  to  it  in 
this  manner.  The  stress  in  a  field  betAveen  two  unlike 
poles  would  drive  a  N-pole  along  the  lines  of  force  in- 
dicated in  Fig.  60  away  from  the  N-pole  toward  the 


EFFECT  OF  HEAT  ON  A  MAGNET.       93 

S-pole.  A  N-pole  placed  between  the  two  N-poles  in 
Fig.  61  halfway  between  them  would  be  in  unstable 
equilibrium,  but  once  started  toward  either  magnet 
would  move  along  the  lines  of  stress  to  the  S-pole  of 
that  magnet  toward  which  it  started. 

The  action  between  two  magnetic  poles  could  be 
explained  by  supposing  the  lines  of  force  to  be  stretched 
elastic  cords  which  are  all  the  time  trying  to  contract, 
and  which  constantly  repel  each  other.  A  strong  field 
of  force  would  be  one  in  which  these  lines  lie  very 
close  together.  In  a  field  of  unit  strength  there  is 
assumed  to  be  one  line  parsing  through  every  square 
centimetre  of  a  plane  drawn  at  right  angles  to  the 
lines.  It  will  be  seen  that  the  lines  may  be  thought  of 
in  two  ways,  either  as  mere  geometrical  lines  indicating 
by  their  direction  the  direction  of  the  force  at  every 
point,  or  as  physical  cords  which  tend  to  shorten  and 
widen  as  a  stretched  rubber  cord  does.  The  direction 
of  the  lines  is  always  supposed  to  be  from  a  north  to  a 
south  pole.  Every  line  is  a  closed  curve  passing  in  at 
one  end  of  the  magnet  and  out  at  the  other. 

67.  Effect  of  Heat  on  a  Magnet. — A  steel  needle 
which  has  been  magnetized  will  lose  its  magnetism  if 
heated  red  hot.  This  is  easily  explained  if  we  remem- 
ber that  the  molecules  tend  to  arrange  themselves  pro- 
miscuously. The  agitation  due  to  heat  sets  them  free, 
they  return  to  their  natural  positions,  and  the  needle  is 
demagnetized.  Jarring  a  magnet  has  a  similar  effect; 


94  ELECTRICITY  AND  MAGNETISM. 

68.  Magnetism    of    the    Earth.  —  The    earth    acts 
like  a  great  magnet,  having  its  poles  at  some  distance 
from  the  geographical  poles,  the'  one  in  the    northern 
hemisphere   being  in  the    northern   part   of    Hudson's 
Bay. 

The  magnetic  needle  points  due  north  at  places  on  a 
line  which  coincides  roughly  with  the  meridian  which 
passes  through  that  pole.  At  places  in  the  Eastern 
States  the  needle  points  west  of  north.  At  places  west 
of  Central  Ohio  the  needle  points  east  of  north.  This 
declination  must  be  known  for  any  place  before  the 
compass  can  be  used  to  tell  directions  accurately.  In 
cities  the  compass  is  now  of  little  use  in  surveying 
because  of  the  presence  of  large  masses  of  iron,  such 
as  gas  and  water  pipes. 

The  horizontal  declination  of  the  needle,  that  is,  its 
variation  from  true  north  in  different  parts  of  the 
world,  was  known  to  Columbus,  who  recorded  the  decli- 
nation for  a  large  number  of  localities  in  the  Atlantic 
Ocean. 

Compass  needles  are  usually  balanced  so  as  to  swing 
only  in  a  horizontal  plane. ;  If  a  needle  is  allowed  to 
point  in  the  direction  of  the  lines  of  force  of  the 
earth's  field,  it  will  dip  downward  at  the  north  end  in 
the  northern  hemisphere. 

69.  Electric    Charges.     Attractions    and    Repul- 
sions. —  It   was   known   to   the    ancient   Greeks   that 
amber,  a  fossil  gum,  when  rubbed,  would  attract  light 


ELECTRIC  CHABGES. 


95 


bodies  like  bits  of  wood  or  pith.  The  Greek  word  for 
amber  —  electron  —  has  given  us  our  words  electrify, 
electricity,  and  so  forth.  A  body  which  has  by  contact 
with  another  body  of  different  material  acq'uired  the 
power  to  attract  light  bodies  is  said  to  be  electrified. 
If  a  glass  rod  be  rubbed  with  silk  the  rod  will  attract 
bits  of  pith,  paper,  and 
other  light  materials.  The 
particles  of  pith  cling  to 
the  rod  for  a  time,  then 
jump  away  as  if  repelled. 
A  light  ball  of  pith  or  cork 
(see  Fig.  63)  hung  by  a  silk 
thread  from  an  insulating 
support  will  be  attracted  to 
the  rod,  and  after  contact 
will  be  repelled.  The  ball 
will  be  attracted  to  a  stick 
and  will  attract  small  par- 
ticles of  pith.  The  ball 
has  become  electrified  by 
touching  the  electrified  rod. 
When  the  rod  was  rubbed  with  the  silk,  the  silk  was 
electrified  as  well  as  the  rod.  The  ball  which  has  been 
charged  by  touching  the  glass  rod  will  be  attracted  by 
the'  silk  but  repelled  by  the  rod.  We  say,  therefore, 
that  there  are  two  kinds  of  electrification,  and  we  call 
the  kind  on  the  glass  positive  or  plus,  that  on  the  silk 
negative  or  minus.  When  a  vulcanite  rod  or  a  bar  of 


FIG.  63. 


96 


ELECTEICITY  AND  MAGNETISM. 


sealing  wax  is  rubbed  with  flannel  or  fur,  the  rubber  or 
wax  is  negatively  electrified,  the  flannel  or  fur  posi- 
tively. Bodies  having  like  electrification  repel,  those 
having  unlike  attract.  Fig.  64  shows  diagrammatically 
the  action  between  pairs  of  rods,  one  of  which  is  elec- 
trified and  suspended  in  a  sling  of  paper  so  that  it  may 
swing,  while  the  other  is  electrified  and  held  near  it. 
The  black  rods  are  vulcanite,  the  light  ones  glass. 


FIG.  64. 

70.  Conductors.  Insulators.  —  Dr.  Gilbert,  physi- 
cian to  Queen  Elizabeth,  examined  a  large  number  of 
substances  and  found  that  under  the  same  conditions 
certain  substances  could  be  electrified,  others  apparently 
could  not.  He  called  them  "  electrics  "  or  "  non-elec- 
trics," according  to  his  results.  Most  of  his  work  has 
proved  correct ;  but,  like  all  men  of  science,  he  taught  us 
also  by  his  mistakes.  He  tried  to  electrify  a  metal  rod 
while  holding  it  in  his  hand,  and  failing,  he  concluded 
metals  were  "  non-electrics."  If  he  had  held  the  metal 
rod  in  a  pad  of  silk  he  would  have  obtained  an  oppo- 


CONDUCTORS.    INSULATORS. 


97 


site  result.  His  facts  were  correct,  for  the  body  carried 
the  charge  to  earth  from  the  metal  rod,  but  his  theory 
was  wrong  because  he  did  not  reckon  in  the  effect  of 
his  hand  on  the  metal  rod.  We  now  classify  bodies  as 
conductors  and  non-conductors  or  insulators.  A  charge 
on  one  part  of  any  body  which  is  an  insulator  does  not 
spread  to  other  parts  of  the  body.  A  charge  on  a  con- 
ductor spreads  in- 
stantly to  all  parts 
of  the  conductor. 
The  metals  are 
good  conductors, 
as  also  is  water- 
vapor,  wet  cotton, 
wet  wood,  moist 
air,  or  any  wet  sub- 
stance.  On  the 
ather  hand,  dry 
cotton,  dry  wood, 
and  dry  air  are 
good  insulators, 
that  is,  poor  con- 
ductors. The  earth  is  a  good  conductor,  so  then  any 
charged  body  which  is  connected  to  the  earth  by  another 
conductor,  such  as  a  wire,  a  gas  pipe,  or  the  human  body, 
will  at  once  lose  its  charge.  We  find  it  necessary,  there- 
fore, when  we  wish  to  charge  a  conductor,  to  insulate  it 
from  the  earth  by  placing  it  on  a  support  of  glass, 
rubber,  paraffine,  varnished  wood,  or  the  like. 


FIG.  65. 


ELECTEIC1TJ  AND  MAGNETISM. 


71.  Electrostatic    Field    of    Force.     Induction. — 

The  charged  body  has  about  it  a  field  of  force  some- 
what similar  to  the  field  about  a  magnet,  but  with  some 
very  important  differences.  The  lines  of  electric  force 
do  not  return  to  the  body  from  which  they  start,  but 
always  end  on  another  body  having  a  charge  of  the 
opposite  kind,  as  shown  in  Fig.  65. 

Any  body  placed  in  a  field  of  force  will  intercept  the 

lines  of  force.  Thus 
a  ball  between  two 
oppositely  charged 
rods  will  be  the  stop- 
ping place  for  lines 
leaving  the  -f-  rod 
and  the  starting  place 


FIG.  66. 


for  lines  going  to 
t  h  e  -  -  r  o  d.  1 1  i  s, 
therefore,  positively 
charged  on  one  side 
and  negative  1  y 
charged  on  the  other 
(see  Fig.  66). 
If  there  is  a  single  charged  body  in  a  room  the  lines 
of  force  go  to  the  walls  of  the  room  or  to  any  object 
which  may  be  near  the  charged  body.  Since  every  line 
has  an  end  as  well  as  a  beginning,  there  is  for  every  pos- 
itive charge  an  equal  negative  charge  somewhere,  which 
is  equivalent  to  saying  that  the  sum  of  all  the  charges 
is  zero.  The  force  between  two  unit  charges  at  1  cm. 


ELECTROSTATIC  FIELD   OF  FORCE. 


99 


distance  is  1  dyne  ;  the  force  between  charges  q  and  q* 
at  distance  r  is,  in  air : 

an)  /=«! 

In  any  other  medium  the  force  is 


where  K  is  a  constant  which  must  be  determined  for 
each  substance.  The  constant  is  called  the  specific  in- 
ductive capacity.  It  is  a  striking  coincidence  that  the 
formulae  representing  the  force  in  the  cases  of  gravita- 
tion, magnetism,  and  electrification  should  be  identical. 
Let  us  see  if  we  can  explain  the  law  in  the  last  case  by 
way  of  example. 

The  field  of  force  at  any  point  is  measured  by  the 
number  of  lines  of  force  to  the  sq.  cm.  at  that  point. 
The  force  at  1  cm.  from  unit  charge  is  1  dyne.  At  the 
same  distance  from  a  charge  of  144  there  are  144  lines  to 


FIG.  67. 


the  sq.  cm.  (see  Fig.  67).  A  body  1  cm.  square  would 
intercept  144  lines  at  1  cm.,  but  it  would  intercept  only 
36  at  2  cm.  and  4  at  6  cm.  For  the  lines  are  twice  as 


100  ELECTEICITT  AND  MAGNETISM. 

far  apart  at  2  cm.  as  at  1  cm.,  hence  there  are  6  rows  of 
6  each  while  at  6  cm.  there  are  but  2  rows  of  2  each. 
The  stress  in  the  field  is  therefore  seen  to  diminish  with 
the  square  of  the  distance.  It  is  obvious  that  to  double 
the  charge  at  A  would  double  the  number  of  lines  of 
force  at  all  points  of  the  field,  or  the  attraction  is  pro 
portional  to  q.  The  same  is  true  of  gf9  hence  the  force 
varies  directly  as  the  product  of  the  charges  and  in- 
versely as  the  square  of  the  distance  between  them. 

72.  Nature  of  the  Electric  Charge.  —  Many,  if  not 
most,  chemical  elements  consist  of  molecules  containing 
two  atoms  which  are  alike  except  that  one  is  electrically 
positive,  while  the  other  is  negative.  When  these  two 
atoms  are  united  in  a  molecule  they  neutralize  each  other 
as  far  as  any  action  outside  the  molecule  is  concerned, 
hence  most  bodies  as  we  find  them  are  not  electrified. 
Let  us  suppose  now  that  bodies  differ  among  themselves 
in  their  affinities  for  the  two  kinds  of  atoms ;  thus  wax 
has  an  affinity  for  negative  atoms,  while  glass  has  an 
affinity  for  positive  ones.  The  oxygen  of  the  air  is 
made  up  of  such  positive  and  negative  atoms.  When  it 
touches  wax  the  wax  is  not  able  to  overcome  the  affinity 
of  the  negative  oxygen  atoms  for  the  positive  atoms  with 
which  they  are  united.  If,  however,  a  piece  of  wax  and 
a  piece  of  glass  are  brought  together  so  close  that  the 
molecules  of  oxygen  in  the  thin  layer  of  air  are  very 
near  to  both  substances  at  the  same  time,  the  plus  atoms 
will  be  drawn  in  one  direction  and  the  minus  atoms  in  the 


NATURE   OF  THE  ELECTRIC  CHARGE. 


101 


opposite  direction ;  their  bond  of  union  is  overcome  and 
the  glass  is  coated  with  plus  oxygen  atoms,  while  the  wax 
is  coated  with  the  same  number  of  minus  oxygen  atoms 
(see  Fig.  68).  The  charge  of  each  atom  being  equal, 
the  two  charges  are  equal  also.  In  our  experiments  we 
use  for  one  of  the  bodies  to  be  electrified  a  silk  cloth 
rather  than  a  bit  of  wax,  because  the  cloth  may  be 


FIG.  68. 

brought  near  to  the  glass  at  many  places  at  once.  We 
also  rub  the  cloth  along  the  rod  to  bring  it  in  contact  at 
as  many  points  as  possible. 

The  physical  lines  of  force,  according  to  the  explana- 
tion just  given,  always  begin  upon  a  negative  atom  and 
end  upon  a  positive  atom.  They  may  be  thought  of  as 
chemical  bonds  stretched  out  to  sensible  distances. 


102 


ELECTRICITY  AND   MAGNETISM. 


73.  The  Electrophorus.  —  A  metal  plate  filled  with 
wax  is  rubbed  with  fur  and  thus  negatively  electrified. 
If,  now,  a  metal  plate  provided  with  an  insulating  handle 
be  brought  near  the  wax  (it  may  touch  it  at  a  few  points 
without  removing  more  than  a  small  fraction  of  the 


FIG. 


charge  from  so  poor  a  conductor  as  wax),  the  plate  will 
have  a  plus  charge  induced  upon  its  lower  surface  (see 
Fig.  69)  and  an  equal  minus  charge,  upon  its  upper 
surface.  The  lines  of  force,  which  start  at  the  upper 
surface,  end  on  the  hand  of  the  operator.  If,  now,  the 
finger  be  touched  to  the  plate  the  negative  charge  will 


DISTEIBUT10N  OF  THE   CHAEGE.  103 

distribute  itself  over  the  body  of  the  operator  and  thence 
to  the  earth,  which  is  so  large  a  conductor  that  the 
charge  at  any  one  place  upon  it  is  practically  zero. 
Meanwhile  the  plus  charge  on  the  lower  side  of  the  plate 
is  held  there  by  the  attraction  of  the  minus  charge  on 
the  wax,  but  if  we  take  away  the  finger  and  lift  the 
plate  to  a  distance  from  the  wax  the  plus  charge  will  dis- 
tribute itself  over  the  plate  and  may  be  drawn  from 
it  in  the  form  of  a  spark  by  presenting  the  knuckle  to 
the  edge  of  the  plate.  This  operation  may  be  repeated 
a  large  number  of  times.  If  the  plate  is  allowed  to  give 
its  charge  to  an  insulated  conductor,  the  latter  may  be 
charged. 

The  charge  on  the  plate  of  the  electrophorus  was  pro- 
duced by  induction. 

74.  Distribution  of  the  Charge  on  a  Conductor.  - 

We  have  said  that  a  charge  distributes  itself  over  a  con- 
ductor. It  does  not  follow  that  it  always  distributes 
itself  uniformly,  however.  This  is  the  case  only  when 
the  conductor  is  a  sphere.  The  farther  the  body  de- 
parts from  the  spherical  form,  the  more  irregular  is  the 
distribution  of  the  charge  on  the  body.  To  express  it 
differently :  the  charge  has  greatest  intensity  at  those 
points  where  the  curvature  is  greatest.  This  is  anal- 
ogous to  the  distribution  of  surface  tension  in  a  soap 
film  and  goes  to  support  the  theory  that  the  charge 
consists  of  a  strain  in  the  ether  at  the  surface  of  the 
conductor.  It  is  evident  that  the  charge  on  a  plate 


104 


ELECTRICITY  AND   MAGNETISM. 


having  rounded  edges  will  be  most  intense  at  the  edges, 
for  there  the  curvature  is  greatest.  If  a  rubber  bag 
were  stretched  over  a  sphere  the  tension  on  the  rubber 
would  be  alike  at  all  points.  If  a  cube  of  the  same  area 
were  substituted  for  the  sphere  the  tension  would  be 
greater  at  the  edges  than  on  the  sides  and  would  be 
greatest  at  the  corners. 

75.  Electric   Discharge. — Any   elastic   medium,    if 

stretched  be- 
yond a  certain 
point,  gives 
way.  The 
air  which 


FIG*  70* 


K 

charged  body 
is  in  a  state 
of  strain. 
When  a  cer- 
tain limit  is 
reached  the 
air  gives  way 

and  the  body  is  discharged  :  equilibrium  is  restored. 
When  the  edge  of  the  charged  plate  of  the  electrophorus 
was  brought  near  the  sphere,  a  spark  was  seen  and 
heard  between  the  plate  and  the  sphere  :  the  plate  was 
discharged.  %  The  air  near  the  sphere  must  have  been 
under  greater  strain  than  at  other  parts  of  the  plate. 
Let  us  see  why,  The  plus  charge  on  the  plate  induced 


ELECTRIC  DISCHARGE.  105 

a  minus  charge  on  the  side  of  the  sphere  nearest  to  it 
(Fig.  70,  £)  and  a  plus  charge  on  the  opposite  side. 
The  minus  charge  on  the  sphere  attracted  the  plus  on 
the  plate,  causing  it  to  accumulate  at  a  until  the  attrac- 
tion of  the  free  plus  atoms  near  the  plate  for  the  minus 
atoms  of  the  oxygen  in  the  air,  assisted  by  the  attrac- 
tion of  the  minus  atoms  near  the  sphere  for  the  plus 
oxj^gen  atoms,  became  great  enough  to  break  down 
the  bonds  uniting  the  oxygen  molecules  in  the  space 
between.  The  result  was  to  produce  equilibrium  be- 
tween the  sphere  and  plate,  but  not  between  these  con- 
ductors and  the  earth/  The  plus  charge  at  c  still 
remains  on  the  ball  and  a  plus  charge  (less  than  the 
original  charge  by  the  amount  on  the  sphere)  remains 
on  the  plate.  If  the  sphere  had  originally  been  con- 
nected to  earth  the  plus  charge  would  not  have  remained 
at  b  and  the  discharge  would  have  resulted  in  perfect 
equilibrium.  The  union  of  the  atoms  of  oxygen  liber- 
ated heat  just  as  the  union  of  oxygen  atoms  with  carbon 
atoms  does  in  combustion.  The  rapid  expansion  of  the 
air  so  suddenly  heated  produced  a  sound  like  that  of  a 
firecracker,  when  powder  burns  suddenly  in  a  confined 
space.  Thunder  and  lightning  are  produced  by  enor- 
mous discharges  from  cloud  to  cloud  in  the  sky.  When 
a  highly  charged  cloud  lies  near  the  earth  it  may  induce 
an  opposite  charge  in  the  earth,  resulting  in  a  discharge 
between  the  cloud  and  the  earth.  The  lightning  strikes, 
as  we  say.  When  lightning  strikes  it  follows  the  best 
conductor.  The  passage  of  a  discharge  through  a  tree 


106  ELECTRICITY  AND   MAGNETISM. 

may  heat  the  sap  enough  to  vaporize  it  and  rend  the 
tree  in  splinters.  The  discharge  from  cloud  to  cloud 
evidently  finds  the  shortest  path  not  always  the  easiest. 
Fig.  71  is  reproduced  from  a  photograph  taken  at  Tripp, 
South  Dakota,  July  22,  1898,  by  W.  C.  Gibbon. 


FIG.  71. 


76.  Discharge  from  Points.  —  The  curvature  at  a 
point  is  infinite ;  the  intensity  at  a  point  is  therefore  so 
great  that  the  charge  escapes  rapidly  into  the  air.  All 
conductors  which  are  designed  to  hold  a  charge  must 
have  their  edges  rounded.  The  discharge  from  the 
points  of  leaves  and  blades  of  grass  is  usually  sum- 


ELECTRICAL   MACHINES. 


107 


cient  to  equalize  the  potential  between  the  earth  and  a 
thundercloud  unless  the  cloud  approaches  very  rapidly. 
This  explains  why  lightning  strokes  are  of  such  infre- 
quent occurrence. 

77.  Electrical  Machines.  — It  is  often  convenient  to 
have  at  our  command  larger  charges  than  can  be  ob- 
tained by  rubbing  a  glass  rod  or  by  means  of  the  elec- 


-tlB 


FIG.  72. 


trophorus.  Machines  employing  the  principles  of  friction 
or  of  induction  are  known  respectively  as  frictional  or 
induction  machines.  An  old-fashioned  frictional  ma- 
chine is  shown  in  Fig.  72.  A  large  glass  plate,  6r,  is 
made  to  revolve  between  two  leather  pads,  P.  The  plus 
charge  on  the  glass  is  carried  past  the  pointed  conductor, 
where  it  draws  off  the  induced  negative  charge,  leaving 


108 


ELECTRICITY  AND   MAGNETISM. 


A  positively  charged.  The  negative  charge  passes  from 
the  pads  to  B>  whence  it  may  be  conveyed  to  earth  by 
a  chain  when  only  a  positive  charge  is  wanted.  This 
machine  has  given  place  to  a  type  of  induction  machines 
in  which  the  inducing  charge  goes  on  increasing  as  the 
machine  is  operated.  Such  machines  produce  much 
more  powerful  charges  than  the  frictional  machine.  The 


FIG.  73. 

kinds  most  in  use  are  the  Toepler-Holtz  machine  and  the 
Wimshurst  machine.  The  Toepler-Holtz  machine  is 
shown  in  Fig.  73  and  diagrammatically  in  Fig.  74.  It 
consists  of  a  revolving  plate,  P,  which  carries  upon  its 
front  surface  six  or  eight  metallic  buttons,  b.  Behind 
the  revolving  plate  is  a  stationary  plate,  P',  which  serves 
to  support  two  paper  armatures,  A,  Af,  as  they  are 
called.  In  front  of  the  revolving  plate  are  an  inducing 


ELECTRICAL  MACHINES. 


109 


bar,  BBf,  with  points  and  brushes  of  wire  at  its  ends, 
and  two  conductors,  (7,  C",  for  receiving  the  charge.  Two 
little  arms,  #,  a',  connected  with  the  armatures,  A,  A', 
convey,  by  means  of  little  metallic  brushes,  any  charge 


FIG.  74. 

that  is  on  the  buttons,  £,  to  the  armatures.  The  action 
of  the  machine  may  be  explained  as  follows :  a  small 
minus  charge  is  produced  on  one  of  the  paper  armatures, 
say  A',  by  rubbing  it  with  flannel.  This  initial  charge 
may  be  exceedingly  small;  indeed,  if  the  machine  has 


110  ELECTRICITY  AND  MAGNETISM. 

been  lately  in  use  the  small  charge  remaining  on  the 
armature  will  be  sufficient.  This  small  charge  acts 
inductively  through  the  glass  on  BB'  to  attract  a  plus 
charge  to  B'  and  repel  a  minus  charge  to  B.  These  in- 
duced charges  will  escape  from  the  points  at  B  and  B', 
electrifying  the  glass.  Part  of  the  charge,  however,  is 
given  to  the  little  buttons,  5,  which,  as  the  plate  rotates, 
carry  it  to  the  brushes  on  a,  ar,  and  thus  to  A,  A'.  The 
charges  on  the  armatures  are  thus  being  continually 
augmented,  and  that,  too,  very  rapidly,  since  the  greater 
the  charge  on  J.,  the  greater  the  induction  on  B.  Mean- 
while the  plus  charge  on  the  upper  part  of  the  glass  plate, 
P,  induces  a  minus  charge  on  the  points  of  (7,  which 
escapes,  leaving  C  positively  charged;  and  the  minus 
charge  on  the  lower  part  of  P  in  like  manner  charges 
0'  negatively.  The  charges  C  and  O1  continue  to 
increase  till  the  air  between  them  gives  way  and  a  dis- 
charge occurs.  The  armatures,  however,  do  not  dis- 
charge, hence  a  second  large  charge  on  C  and  0'  may 
be  obtained  quickly.  Indeed,  if  C  and  O'  are  not  too 
far  apart,  an  almost  continuous  succession  of  sparks  may 
be  made  to  pass  between  them. 

78.  Potential.  Capacity.  —  If  a  ball  of  1  cm.  radius 
receive  unit  charge  it  will  be  more  intensely  electrified 
than  a  ball  of  10  cm.  radius  having  the  same  charge, 
exactly  as  a  given  quantity  of  heat  applied  to  the  small 
ball  will  heat  it  hotter  than  an  equal  quantity  applied 
to  the  large  one.  The  term  in  electricity  which  corre- 


POTENTIAL.     CAPACITY.  Ill 

sponds  to  temperature  in  heat  is  potential.  The  poten- 
tial of  a  ball  of  1  cm.  radius  having  a  charge  of  plus  1 
unit  is  1 ;  the  potential  of  the  ball  of  10  cm.  radius 
having  the  same  charge  is  0.1.  The  capacity  of  a  con- 
ductor is  the  quotient  of  its  charge  by  its  potential. 
The  capacity  of  a  body  is  known,  then,  if  we  can  meas- 
ure the  charge  which  it  receives,  from  a  source  of 
known  potential.  Potential  is  usually  denoted  by  V. 
If  we  denote  quantity  of  charge  by  Q  and  capacity  by 
0  we  may  write  : 

(13)     0=  Q- 
V 

The  potential  of  the  earth  is  reckoned  as  zero.  The 
potential  of  all  conductors  in  contact  with  each  other 
soon  becomes  identical,  just  as  the  temperature  of  two 
bodies  in  contact  soon  becomes  the  same. 

The  potential  of  a  conductor  which  is  connected  to 
the  earth  is  therefore  zero.  The  potential  of  ihe  space 
surrounding  a  body  diminishes  as  we  leave  the  body. 
A  charge  always  "  flows  "  from  places  of  high  to  places 
of  low  potential,  just  as  heat  flows  from  places  of  high 
to  places  of  low  temperature.  A  discharge  which  takes 
place  through  a  good  conductor,  like  a  copper  wire,  is 
usually  called  a  current.  We  are  not  to  infer  from  the 
use  of  the  term  current  that  we  are  dealing  with  a  fluid. 
What  takes  place  in  the  wire  seems  to  be  different  from 
what  takes  place  in  the  air  when  a  spark  passes,  but  the 
result  accomplished  is  the  same  :  equilibrium  or  equal- 
ity of  potential  is  restored  in  both  cases. 


112 


ELECTRICITY  AND  MAGNETISM. 


FIG.  75. 


79.  Condensers.  —  The  potential  of  the  space  about 
a  conductor  is  lowered  by  the  presence  of  a  conductor 
of  lower  potential.  A  plate,  A  (Fig.  75),  connected  to 
the  positive  conductor  of  an  electrical  machine,  (7,  having 
a  potential  of  100  will  itself  have  a  potential  of  100. 

If  the  capacity 
of  the  plate  is 
10  the  plate  will 
receive  a  charge 
of  1,000  units 
before  its  poten- 
tial rises  to  100. 
If  we  now  hang 
near  A  a  plate, 
B,  which  is  connected  to  the  earth,  the  potential  on  the 
side  of  A  next  to  B  will  be  lowered  and  an  additional 
quantity  must  flow  from  0  before  the  potential  of  A 
will  rise  to  100.  The  closer  B  is  to  A  the  greater  the 
charge  required  to  equalize  the  potential. 

Such  an  arrangement  of  plates  as  that  just  described 
is  called  a  condenser  because  it  allows  a  large  charge  to 
be  collected  in  a  small  space,  or,  in  other  words,  it 
increases  the  capacity  of  the  conductor  A.  A  common 
form  of  condenser  is  the  Leyden  jar.  It  consists  of  a 
glass  jar  coated  to  about  two  thirds  its  height  with  tin- 
foil, both  outside  and  inside.  A  rod  terminating  in  a 
ball  passes  through  the  insulating  cover  and  communi 
cates  by  a  chain  to  the  inner  coating  (see  Fig.  76).  The 
glass  has  a  specific  inductive  capacity  six  times  as  great 


113 


as  that  of  air,  hence  the  capacity  is  six  times  as  great 

as  a  condenser  having  plates  of  the  same  size.     The  jar 

is  charged  by  connecting  the  ball  to  one 

pole  of  the  electrical  machine  while  the 

outer   coating   is    held    in    the    hand    or 

otherwise   connected    to    earth.     It   may 

be  discharged  by  touching  one  end  of  a 

wire  having  rounded  ends  to   the   outer 

coating  and  then  bringing  the  other  end 

near  the  knob.     A  very  powerful  charge 

will  sometimes  pierce  the  jar.     Franklin 

devised  an  experiment  to  prove  that  the 

charge   is   wholly  in    the    dielectric    and 

not  at  all  in   the    conductor.     A    jar   is  FI<*- 

made  with  removable  coats  (see  Fig.  77).     After  being 

charged,  the  inner  coating  is  removed  by  means  of  a  glass 

rod  or  its  ebonite  handle,  and  the  jar  is  lifted  out  of 


FIG.  77. 


the  outer  coating.  The  coatings  now  show  no  signs  of 
electrification,  while  the  jar  attracts  pith  balls.  If  the 
coatings  are  again  put  in  place  the  jar  will  give  a  spark. 


114  ELECTRICITY  AND  MAGNETISM. 

The  purpose  of  the  coatings  is  to  distribute  the  charge 
to  the  glass  in  charging  and  to  collect  it  at  the  moment 
of  discharge.  That  the  charge  is  not  all  given  up  by 
the  glass  is  shown  by  the  fact  that  it  is  usually  possible 
to  get  a  second  (or  residual)  charge  a  few  moments 
after  a  condenser  has  been  discharged. 

Electrical  machines  are  usually  provided  with  two 
Leyden  jars.  The  positive  conductor  is  connected  with 
the  inner  coating  of  one  jar,  the  negative  with  the 
inner  coating  of  the  other,  and  the  two  outer  coatings 
being  connected  by  a  wire  which  may  be  broken  by  a 
switch  when  it  is  not  desired  to  use  the  condensers 
(see  Fig.  73). 

80.  The  Gold-leaf  Electroscope.  —  It  is  often  desir- 
able to  make  a  more  delicate  test  of  electrical  charges 
than  can  be  made  with  the  pith  ball.  For  this  purpose 
a  gold-leaf  electroscope  is  used.  It  consists  of  a  rod 
rounded  or  furnished  with  a  ball  at  its  upper  end  and 
having  attached  to  its  lower  end  two  strips  of  gold-foil. 
The  rod  passes  through  the  stopper  of  a  bottle  which 
encloses  the  leaves,  protecting  them  from  draughts  of 
air  (see  Fig.  78).  When  a  glass  rod  is  brought  near 
the  ball  the  electroscope  has  a  positive  charge  induced 
in  the  leaves  by  induction.  If  while  the  rod  is  near 
the  finger  is  touched  to  the  knob  the  repelled  plus  charge 
will  pass  to  earth,  leaving  the  electroscope  negatively 
charged.  If  we  remove  the  finger  and  then  the  rod 
the  electroscope  will  have  a  permanent  charge.  A  nega- 


THE  GOLD-LEAF  ELECTROSCOPE. 


115 


tively  charged  body  brought  near  will  cause  the  leaves 
to  diverge  further,  while  a  positively  charged  body  will 
cause  the  leaves  to  collapse.  If  a  rubber  rod  had  been 
used  instead  of  the 
glass  rod  the  electro- 
scope would  have 
been  positively 
charged.  A  conven- 
ient method  for  test- 
ing bodies  for  kind 
and  amount  of  elec- 
trification is  to  use 
what  is  called  a  proof 
plane.  It  consists  of 
a  small  metal  button  • 

r  I  (jr.      Jo* 

with  round   edges 

cemented  to  the  end  of  a  glass  rod.  A  metal  ball  or 
button  tied  to  a  silk  thread  will  do  as  well.  The  proof 
plane  is  touched  to  the  body  to  be  tested  and  the  charge 
received  by  the  plane  is  carried  to  the  electroscope.  If 
the  charge  is  not  strong  the  plane  may  be  touched  to 
the  ball  of  the  electroscope. 

By  means  of  an  electroscope  Faraday  proved  that 
the  charge  on  a  conductor  induced  by  a  charged  body 
brought  near  it  is  equal  to  the  charge  on  the  inducing 
body.  He  performed  the  experiment  with  an  ice-pail, 
hence  it  is  known  as  Faraday's  ice-pail  experiment.  It 
is  an  excellent  illustration  of  the  laws  of  induced 
charges.  A  small  pail,  P  (Fig.  78),  is  supported  on  an 


116  ELECTRICITY  AND   MAGNETISM. 

insulated  support  and  a  ball  which  has  been  charged  by 
contact  with  a  rubber  rod  is  lowered  by  a  silk  thread 
into  the  pail.  The  leaves  of  the  electroscope  will 
diverge  a  certain  amount.  If  the  ball  be  now  touched 
to  the  pail  the  charge  on  the  ball  will  just  neutralize 
the  induced  charge  on  the  pail  and  the  divergence  of 
the  leaves  will  not  be  in  the  least  altered.  This  shows 
that  the  charge  induced  on  the  pail  was  exactly  equal 
and  opposite  to  the  inducing  charge  on  the  ball. 

81.  The  Discharge  in  Gases.  —  If  we  draw  apart  the 
knobs  of  an  electrical  machine  while  it  is  in  operation  a 
point  will  be  reached  at  which  sparks  can  no  longer 
pass,  but  a  hissing  noise  may  be  heard.  If  the  room  is 
darkened  we  may  see  faint  brushes  of  purplish  light 
issuing  from  one  of  the  knobs.  It  will  be  found  by 
testing  that  this  is  the  negative  conductor.  Similar 
brush  discharges  may  be  seen  on  the  plate  and  at  the 
pointed  conductors  on  various  parts  of  the  machine. 
The  positively  charged  points  show  little  dots  of  white 
light  instead  of  the  brushes.  If  we  now  connect  the 
two  knobs  to  wires  which  are  sealed  into  the  end  of  a 
glass  tube  which  is  connected  to  an  air  pump,  it  will  be 
observed  that  when  part  of  the  air  has  been  exhausted 
from  the  tube  brush  discharges  will  pass  through  the 
tube,  and,  if  the  exhaustion  can  be  carried  far  enough, 
the  whole  tube  will  glow  with  pink  light.  Tubes  may 
be  obtained  from  which  the  air  has  been  exhausted 
until  not  more  than  the  thousandth  part  remains.  They 


EXEECISES.  117 

are  called  Geissler's  tubes.  In  such  tubes  the  negative 
end  glows  with  a  violet  light,  while  the  positive  end  is 
pink,  often  arranged  in  layers  or  stratifications  concave 
toward  that  end  and  reaching  nearly  to  the  negative 
end  of  the  tube.  A  dark  space  separates  the  two  dis- 
charges. The  colors  are  not  the  same  if  other  gases 
are  in  the  tubes.  The  discharge  in  such  exhausted 
tubes  bears  a  close  resemblance  to  the  aurora  borealis  or 
northern  lights,  which  are  believed  to  consist  of  electri- 
cal discharges  through  the  rare  air  in  the  upper  regions 
of  our  atmosphere. 

Tubes  which  are  exhausted  to  a  still  greater  degree 
than  Geissler's  are  known  as  Crookes'  tubes.  The  slight 
amount  of  gas  remaining  in  a  Crookes'  tube  shows  no 
color,  but  the  tube  itself  glows  with  a  phosphorescent 
light  opposite  the  negative  end.  The  tubes  are  the 
source  of  Roentgen  X-rays,  which  will  be  considered  in 
another  place. 

Exercises. 

52.  (a)  Place  a  bar  magnet  on  a  large  sheet  of  paper.     Set 
a  small  compass  on  the  paper  and  make  a  dot  on  the  paper  near 
each  end  of  the  needle.     Set  the  compass  aside  and  connect 
the  dots  by  a  line.     Kepeat  this  operation  for  a  large  number 
of  places   on   the   paper.     (&)  Support   a  magnetized   sewing 
needle  by  a  thread  fastened  to  it  with  wax,  so  that  it  will  hang 
horizontally  when  no  magnet   is  near  it.     Bring  it  over  the 
centre  of  a  bar  magnet  at  a  distance  of  say  10  cm.  above  the 
magnet,  and  then  move  it  slowly  toward  one  end  of  the  magnet 
and  afterward  toward  the  other  end. 

53.  A  sailor  was  asked  if  he  ever  crossed  the  "line  "  (equa- 
tor).   He  replied  that  he  had,     When  asked  how  he  knew,  he 


118  ELECTRICITY  AND   MAGNETISM. 

replied  that  he  watched  the  compass  and  the  instant  the  ship 
crossed  the  line  the  needle  turned  and  pointed  south.    Explain. 

54.  An  artesian  well  made  by  driving  a  long  iron  pipe  into 
the  ground  was  found  to  possess  magnetic  properties,  such  that 
knife  blades  thrust  into  the  water  as  it  flowed  from  the  pipe  were 
magnetized.     Is  it  more  probable  that  the  water  was  magnetic 
or  that  the  iron  pipe  was  magnetized  by  being  jarred  while  in 
the  earth's  field  ?     If  the  point  of  a  knife  were  rubbed  on  the 
upper  end  of  this  pipe,  what  sort  of  pole  would  the  point  be- 
come? 

55.  "Why  should  care  be  taken  never  to  drop  or  otherwise  jar 
a  magnet  ? 

56.  Why  are  magnets  often  made  in  a  horseshoe  form  ? 

57.  A  knitting  needle  is  bent  into  a  horseshoe  form  and  then 
magnetized.     If  the  distance  between  its  ends  were  measured 
before   it  was  magnetized  and  again  afterwards,  would  it  be 
found  greater  before  or  after  magnetization  ? 

58.  Would  it  probably  have  any  effect  on  the  time-keeping 
qualities  of  a  watch  if  its  hairspring  should  become  magnetized  ? 


N 


FIG.  79. 

59.  A  long  bar  magnet  has  poles  of  strength  80.     A  compass 
needle  having  poles  of  strength  2  is  pointing  toward  the  X-polo 
of  the  magnet.     The  needle  is  2  cm.  long  and  its  nearest  pole 
is  4  cm.  from  the  N-pole  of  the  magnet  (see  Fig.  79).     What 
is  the  force  of  attraction  in  dynes,  what  is  the  force  of  repul- 
sion, and  what  the  resultant  force  ?     What  would  the  resultant 
force  be  if  the  distance  were  half  as  great  ? 

60.  If  an  iron  rod  be  tapped  with  a  hammer  while  it  points 
in  the  direction  of  the  dip  needle  it  will  become  a  mag-net  with 
the  lower  end  a  N-pole.     If  we  now  reverse  it  and  tap  it  the 


EXEECISES.  119 

magnetism  will  be  reversed,  and  the  end  which  was  at  first 
down  and  is  now  up  is  a  S-pole.  In  what  position  should 
it  be  held  111  order  that  it  may  be  wrholly  demagnetized  by 
tapping  ? 

61.  (a)  Plot  the  field  of  a  horseshoe  magnet  by  means  of 
iron  filings.     (6)  In  a  similar  way  plot  the  field  between  two 
unlike  poles  of  bar  magnets  at  a  distance  of  8  cm.  from  each 
other  and  with  a  piece  of  soft  iron  1  or  2  cm.  square  lying  mid- 
way between  them. 

62.  A  1-milligram  weight  lies  on  a  glass   plate   and  has  a 
charge  of  10  units  upon  it.     How  large  a  charge  at  a  distance 
of  2  cm.  will  lift  the  weight  from  the  glass  ? 

63.  Charge  an  insulated  pail  and  test  it  inside  and  out  with 
a  proof  plane  and  electroscope. 

64.  Put  an  electrical  machine  in  motion  and  test  the  various 
parts  for  positive  and  negative  electrification.     Do  not  let  the 
machine  stop  till  the  test  is  finished.     Stop  the  machine,  start 
again  and  repeat  the  tests  to  see  if  you  get  any  indication  that 
the  machine  reverses. 

65.  Why  is  it  more  difficult  to  brush  lint  from  clothing  in 
cold,  dry  weather  than  at  other  times  ? 

66.  Lay  a  pane  of  glass  on  the  table  and  rub  it  with  a  silk 
cloth.     Dust  a  few  fine  iron  filings  on  the  glass.     Lift  the  glass 
and  note  the  behavior  of  the  filings.     Explain. 

67.  Kub  a  sheet  of  paper  with  a  dry  flannel  cloth  and  place 
it  against  the  wall  of  the  room. 

68.  Let  water  from  an  elevated  vessel  flow  out  of  a  pipette 
at  the  lower  end  of  a  rubber  tube.     Bring  an  electrified  rod 
near  the  water  jet.     Note  and  explain  (a)  the  movements  of 
the  jet ;  (6)  any  changes  in  the  jet  itself. 

69.  Connect  two  metal  handles  by  wires  to  the  outside  coat- 
ings of  the  jars  of  an  electrical  machine  ;  arrange  the  knobs  so 
that  a  short  spark  passes.     By  holding  the  handles  observe  the 
physiological  effect  of  an  electric  current. 


120 


ELECTRICITY  AND   MAGNETISM. 


82.  Electric  Currents.  —  While  electric  charges  are 
of  great  interest  to  us  from  the  theoretical  side,  the 
greater  part  of  the  applications  of  electricity  employ 
what  we  call  electric  currents.  In  1786  Galvani  ob- 
served that  some  freshly  prepared  frogs'  legs  lying  near 
an  electrical  machine  twitched  when  a  discharge  passed. 
In  trying  to  use  frogs'  legs  as  an  electroscope  to  test 
the  electrification  of  the  air  the  frogs'  legs  came  in  con- 
tact with  an  iron  balcony  and  twitched  when  removed. 


FIGS.  80  and  81. 


Galvani  believed  this  was  due  to  a  charge  generated  in 
the  muscles  of  the  specimen  used,  but  Volta  thought  it 
could  be  explained  by  the  contact  of  dissimilar  metals. 
Volta  constructed  the  first  battery  and  from  this  time 
dates  the  study  of  electric  currents.  If  a  strip  of  zinc 
which  is  connected  to  ground  be  touched  by  a  piece  of 
copper  which  is  connected  to  a  very  sensitive  electro- 
scope, the  leaves  of  the  electroscope  will  diverge,  show- 
ing a  difference  of  potential  between  the  two  plates 


ELECTRIC   CURRENTS.  121 

(see  Fig.  80).  This  difference  of  potential  by  contact  of 
dissimilar  substances  has  been  explained  in  Section  72. 
If  we  now  connect  the  wire  which  was 
attached  to  the  earth  to  the  one  attached  to 
the  electroscope  the  charge  will  disappear. 
A  current  has  flowed  through  the  wire  from 
the  copper  to  the  zinc  (see  Fig.  81).  The 
current  lasts  but  an  instant,  however,  like 
the  discharge  from  a  Leyden  jar,  only  much 
weaker.  If  instead  of  bringing  the  metals  FlG>  S2' 
in  contact  in  air  we  bring  both  metals  in  contact  with 
water,  which  has  a  salt  or  an  acid  dissolved  in  it,  the 
current  will  flow  continuously  (see  Fig.  82).  Let  us  try 
to  explain  what  occurs.  The  acid  which  is  dissolved  in 
water  consists,  let  us  say,  of  a  positive  atom  of  hydro- 
gen, the  chemical  symbol  for  which  is  H,  united  to  a 
negative  atom  of  chlorine,  the  symbol  for  which  is  Cl. 
The  molecules  of  acid  in  a  solution  are  not  like  the  mole- 
cules of  oxygen  in  air,  but  a  great  many  of  the 'mole- 
cules are  dissociated  so  that  they  pass  quickly  to  the 
metals,  the  plus  H  going  to  the  copper,  while  the  minus 
Cl  goes  to  the  zinc.  When  the  plates  are  joined  out- 
side  the  solution  by  a  wire  a  discharge  occurs  through 
the  wire,  but  other  atoms  of  H  rush  to  the  copper 
plate  and  other  atoms  of  Cl  to  the  zinc  plate,  so  that 
the  current  is  kept  up  continuously.  The  H  atoms 
unite  with  each  other  and  form  hydrogen  bubbles  which 
rise  to  the  surface  of  the  liquid,  while  the  chlorine 
combines  with  the  zinc. 


/,/,'»•/ /,-/•  r/'i  i \ /'  i/ 1.,  \ /  //  >/ 


M  i     I'olriil/.fiiloii         II"    •  luol'  illlll 
in   ill-    M  II     "I    hull  ......  I'OV       ""'    III    Mi-     Inrl     Mint.    Mm 

I",!,  MUM    nf    KUN   1'imiM'il    nn    Mm   |>lnl.n   .1  ......  1    <|m.  II-. 

ll'lIM'     Mll«     jilllilMl,      I'lll       l.»  ill     .1      I   i\  .    i      ..I        fll        IIVI'I       Mmill, 
\\   III'    li        I'M     \  .    Ml  Ml.         ill       :.H    Illtl    .1        Ml.  -in  I  ........  l|||    Imp- 

Mm     jilnli'M     mill     MII    wi'iiKi'MM    Mn      .  i  .....  >l     nml     liimllv 

ClIIIMl'M     II      III    COIIMO    llllllMMl      I'llhi-   l\  I  I.I",      |  >lirl  ininrllnli 

i       I  M"\\  ii     mi     /'.-An  /  ,iti>»nt     nml     Mm    rhml     Jill  ....... 

l"i\\  .....  (ill  .....  I    MM|'|H   n!'    Imllrl'IrM    Im    in    Mm    vnriniiN 

II     •    -I     IM    H'|l|,    I'M     III'        Ml|l|ll  -III.  .11 


M.j       IVI:ir.iirnr     Mini      ol      Cm  iriil  .-..          OPIM|(M|      ilin 
•I     III     I  SIP    Mud     n     \\  ......  -rylllK    l»    «  nnrnl      \\  ill 

ilrlliM-l  ii  111  i  ;in  III  lii  .'illr,  II' 
Mio  \\  i  iv  IM  ImM  |..ii(illrl  In  Mm 
Itl  •  -II'  M.  ......  Hit  |r|nl',  In  Im  n 

ni  ii-;iii  ingUi  in  MI.  u  n.  ,  bhi 

in.  ..mil    |||    .l.-llri  li.xi    n|     III.     Ml  , 
tllr     .1,   |M   mini.;     ..n      llio     N(lVll-;Mi 

.-i  i  h.    \\  n.       A  IHMM||I-  \\  iiu-li  in 

ii     |"    II.  I.   'I      .'II       i      Imll.'iHllil!     ((MM 


I   .   i     MM 


will,   1  11 


in 


.notMullH  nmun 

in      i     eotloil   --i 


\vlu.l)    IM    PiirrvllIK    u    tMU'lViil.   I  il.. 

III    Kl^,   HI),     Tlu»   tMMilnil  Mpnl 

Mio  \^|iv,      SliH'n    Ilio   I»MM||I»   nl\\  i\  -. 

I"'  Jl<  I    i"   bhl   liii^M  uf    I'oivo  III    HlO    llrl.l.    il.n.'    IUU    i     |  Hi 

ill  Mm  MpMOO  \\  tllt«||  MlimitllulH  II   •  .'ii.  In,  I...    .    .i  .  \  Inj     .    .  HI 

rou!    ti    llohl    nl     nt:i.;imMo    \\\\v\*   in    \\ln.li    ilio    Imt'M  t>f 
4^  ollvK'M  iMH'loHhl;    il"    -  .'ii.hi.  I.M 


A     .  <  m  |  P.I         .  i 

I  1 1  1 1 1 1 1 1 1 ;  •      I  M    1 1 .   .  i  I  1 1      i 
III!'       \\  IM         .    .11  I  \ 

Mi  •• Ill 

IH         ,,.,,,,,,     |,    ,  | 
Ml.ll       M I   M     Ml 

lln\\  ;•   |  HUH    IK. i  I  I, 

i..    <.ui  Ii  I  In., u"l 
MM      !Mi  I  \  i 

•  IIJH      lli«    IM  «  ,11, 


111    ,i    i 
mill-.  CM 


I     ni    \\  i  H  il 
.  mi|.l,    f/it 


•'  i  i  ii    .1 


, ,.    i 


1    '   •''    '' 


|  '   '  •    I'  itf.  H4).     I'   1  1  .....  " 

IM     <  li.ih",  -I      i)    I  li.il     I  I,.     .1111.  nl    ||,,  •       I  I,,     ,,|  I,,  , 

ITftJ       'I  .......  II'     W  iH    •!«  II  ........  II      -  II    I 

II       "  •        l.<   ml       I  Ii.          •   ii.        I,.  i.    I        ,,     .    i        I  I,, 

Nil  Hi-  'I     ll  ......  M  nl    ftoWl   iM'illi 

'I  ......  <H<   ,     l"ll      :  ......  h      IH   |..\V     ||(f 

MM        .1.    II.   ,    hull      U  |||       I,.         ,,,,    M    ..     .   .|,      HJHl'il 

•    "  Ii      |..'H      M|       III.        \\  ii,       .(.  II,  ,  I*      ||,     III 
I  Iii        ..in.     .In.  .  1  1,  111    (  Nd(1    I1  I!'.    >'M'I  )         liy 
Id-          IM      .,      i,u  ml,.  ,      ,,|      I,,,,, 

Mir    111  .  .11,     Mi.     .  iii  .  i     i      .  ni  M 

in-  pi  H  •  •'  »nd  n  .....  iti  ami  ni 

I  .....  "n«N   NMHNhJVn   IMHMI^Il    I"  «!•  l«  •  I    -  III 

rmiU  nl'  VMry  Mimill   in 


>  I0i  NA, 

H5*  Bdtteriei,      A    i.iii.>     '.iiiHiHiM   «^«i'nii(illy   <>r 

WG    Uhlil   .....  nlii.   I,,)       imm.   ,     .  -I     |D    |       .,luli.,i,          <  h,.      .,| 

1)0  con.  in.  ton  Ii  .1  >M<  i.  M,  UK  niii.  r  Noiturtimti  A 


124 


ELECTRICITY  AND   MAGNETISM. 


sometimes  carbon,  carbon  being  almost  the  only  solid 
substance  outside  of  the  metals  which  is  a  good  con- 
ductor. We  have  seen  that  some  plan  must  be  used 
to  get  rid  of  the  products  of  the  chemical  change  which 
occurs  in  the  battery  and  so  avoid  polarization.  The 
batteiy  most  commonly  used  is  the  Daniell's  cell  in  the 
form  known  as  the  gravity  cell,  because  its  two  fluids  are 
kept  apart  by  their  difference  in 
specific  gravity  (see  Fig.  86). 
We  may  begin  with  a  cell  hav- 
ing a  copper  plate  at  the  bot- 
tom, a  zinc  plate  at  the  top,  and 
a  very  dilute  solution  of  sul- 
phuric acid.  Sulphuric  acid 
consists  of  two  radicals  or  com- 
binations of  atoms  which  are 
treated  like  atoms,  namely,  hy- 
drogen and  sulphion,  H2  and 
SO4.  When  the  plates  are  connected  by  a  wire  the  hy- 
drogen collects  on  the  copper  plate  and  soon  polarizes 
the  battery.  To  prevent  this  action  the  copper  plate  is 
covered  with  copper  sulphate,  CuSO4.  Now  sulphion, 
as  the  radical  SO4  is  called,  has  a  stronger  affinity  for 
hydrogen  than  for  copper,  so  it  gives  up  its  copper  atom, 
which  is  deposited  on  the  copper  plate,  and  unites  with 
the  hydrogen  to  form  sulphuric  acid.  The  chemist  ex- 
presses this  reaction  by  an  equation  thus : 


FIG.  86. 


H2  +  CuS04  =  Cu  +  H2S04 


BATTEBIES. 


125 


But  sulphion  has  a  stronger  affinity  for  zinc  than  for 
hydrogen,  so  it  gives  up  its  hydrogen  and  takes  zinc, 
making  zinc  sulphate,  ZnSO4  : 

Zn  +  H2S04  =  H2  +  ZnS04 

and  so  the  rnerry  round  keeps  up  and  the  difference  of 
potential  remains  constant  while  a  constant  current 
flows  through  the  wire.  The 
battery  thus  consumes  zinc  and 
copper  sulphate,  while  it  pro- 
duces copper  and  zinc  sulphate. 
The  zinc  sulphate  must  occa- 
sionally be  siphoned  off  and 
replaced  by  water,  while  cop- 
per sulphate  crystals  are  added 
to  make  good  the  loss.  This 
is  the  only  battery  which  works 
best  if  kept  in  constant  action. 
A  gravity  or  Daniell's  cell 
should  be  kept  on  closed  circuit 
when  not  in  use  ;  all  other  bat- 
teries should  have  the  circuit 
broken  instantly  when  they  are 
not  to  be  longer  used. 

While  the  current  furnished  by  the  gravity  cell  is 
steady,  it  is  not  strong.  The  various  carbon  batteries, 
known  as  Leclanche*  or  sal  ammoniac  batteries,  give 
a  strong  current  for  a  short  time,  but  soon  polarize. 
They  recover  quickly,  however,  and  so  are  well  adapted 


FIG.  87. 


126 


ELECTEICITY  AND   MAGNETISM. 


to  use  for  electric  bells  and  telephones.  Usually  a  zinc 
rod,  a  large  carbon  plate,  and  a  solution  of  ammonium 
chloride  (sal  ammoniac)  constitute  the  battery  (see 
Fig.  87).  The  bubbles  of  hydrogen  soon  escape  from 
the  large  rough  surface  of  the  carbon.  This  battery  is 
cheap  and  easily  renewed. 

When  a  strong  current  for  a  considerable  time  is 
wanted  the  plunge  battery  or  chromic  acid  battery  is  used. 
In  this  battery  a  large  zinc  plate  is  hung  from  the  lower 
side  of  a  piece  of  vulcanite  between  two  carbon  plates. 
The  plates  are  plunged  into  a  solution  of  chromic  acid. 
The  plates  must  be  lifted  whenever  the  battery  is  not 
in  use.  This  battery  requires  frequent  cleaning  and 
renewal,  but  it  is  the  best  resort  for  strong  currents 
when  storage  batteries  or  dynamo  currents  are  not  to 
be  had. 

86.  Chemical  Effects  of  the  Current.    Electrolysis. 

-The  direction  taken 
by  the  dissociated  atoms 
in  a  solution  with  two 
unlike  plates  dipping  in 
it  is  determined  by  the 
nature  of  the  plates.  If 
two  like  plates  dip  in 
such  a  solution  there  will 
be  nothing  to  determine 

the  direction,  and  consequently  no  effect  will  be  ob- 
served. Now  suppose  two  plates  of  platinum,  which 


ELECTBOPLATING.  127 

dip  in  dilute  sulphuric  acid,  are  connected  to  the  plates 
of  a  battery,  as  shown  in  Fig.  88.  Atoms  of  hydrogen 
at  once  go  to  the  negative  ^plate,  the  sulphion  takes  hy- 
drogen from  the  water  molecules  (H2O),  leaving  free 
oxygen,  which  collects  on  the  positive  \plate.  The  final 
result  is  to  decompose  the  water  into  hydrogen  and  oxy- 
gen. If  the  plates  are  held  under  test  tubes  the  gases 
may  be  collected,  when  the  volume  of  hydrogen  will  be 
found  just  twice  that  of  the  oxygen. 

If  lead  plates  had  been  used  the  oxygen  would  have 
united  with  the  lead,  forming  lead  oxide.  The  cell,  with 
two  unlike  plates,  lead  and  lead  oxide,  would,  when  dis- 
connected from  the  battery,  be  itself  a  battery.  This, 
somewhat  modified,  is  the  principle  of  the  lead  storage 
battery. 

87.  Electroplating.  —  An  important  application  of 
electrolysis  to  the  arts  is  in  the  deposition  of  metals  by 
the  electric  current.  The  object  to  be  coated  is  made 
the  negative  terminal  (attached  to  the  zinc  plate)  of  a 
battery,  while  the  positive  terminal  is  composed  of  a 
plate  of  the  metal  to  be  deposited.  Both  plates  dip  in 
a  solution  of  a  salt  of  the  metal  to  be  deposited.  The 
current  deposits  the  metal  on  the  negative  terminal, 
or  electrode,  as  it  is  called,  while  the  metal  from  the 
positive  electrode  goes  into  the  solution. 

It  is  sometimes  easiest  to  separate  metals  from  the 
ores  which  contain  them  by  chemically  producing  the 
salts  of  the  metals.  The  metals  may  then  be  recovered 


128  ELECTRICITY  AND   MAGNETISM. 

from  the  salts  by  electrolysis.  Copper  is  produced  on 
a  large  scale  in  this  way. 

88.  Electrotyping.  —  Books  are  now  seldom  printed 
from  the  movable  type  in  which  the  matter  was  at  first 
set.     A  mould  is  made  from  the  type  in  wax,  which  is 
then  coated  with  finely  powdered  graphite  (a  form  of 
carbon)  to  make  it  conduct,  and  is  made  the  negative 
electrode  in  a  bath  of  copper  sulphate.     The  current 
deposits  copper  in  the  mould  in  a  form  which  perfectly 
reproduces    the    type.     The    thin    sheet    of    copper    is 
backed  by  type  metal  and  wood  of  the  proper  thickness 
and  the  types  are  free  to  be  used  again  without  having 
been  worn  in  the  least,  while  the  plates,  after  use,  may 
be  laid  away  till  another  edition  of  the  book  is  wanted. 

89.  Heat  Produced  by  the  Current.    Resistance.  - 

When  one  body  moves  upon  another  the  rubbing  of  the 
molecules  against  each  other  sets  them  in  motion,  or, 
we  say,  friction  produces  heat.  Something  analogous 
to  friction,  but  of  course  very  different,  hinders  the 
transfer  of  an  electric  charge.  We  have  seen  that  the 
air  is  intensely  heated  in  the  path  of  a  spark  discharge. 
A  wire  offers  much  less  resistance  to  the  passage  of  a 
current  than  air  does,  but  the  current  always  heats  the 
•wire  and  the  fluid  of  the  battery  also.  If  the  fluid  is  a 
good  conductor,  and  a  bad  conductor  is  in  the  circuit 
outside  the  battery,  the  battery  will  be  heated  but  little, 
while  if  the  battery  is  a  poor  conductor  and  the  circuit 


HEAT  PRODUCED  BY  TtiE   CURRENT.          129 

is  a  good  one,  the  battery  will  become  heated  more  than 
the  wire.  The  resistance  of  a  conductor  is  the  exact 
opposite  of  conductivity.  If  we  indicate  conductivity 
by  C  and  resistance  by  R  we  may  write  : 

0  =  —  or  R  =  — 
R  0 

Ohm  proved  that  the  resistance  of  a  conductor  of  uni- 
form cross  section  varies  directly  as  its  length  and  in- 
versely as  its  cross  section.  Two  wires  of  the  same 
length  and  diameter  have  different  resistances  if  they 
are  of  different  materials  ;  that  is  to  say,  substances  have 
specific  resistance  (or  conductivity)  just  as  they  have 
specific  density  and  specific  heat.  The  resistance  of  a 
conductor  whose  uniform  cross  section  is  a,  whose  length 
is  £,  and  whose  specific  resistance  is  r,  is  : 


If  the  cross  section  is  a  circle  the  area  of  the  cross  sec- 
tion is  3.1416  times  the  square  of  the  radius.  The 
cross  section  of  two  wires  of  equal  diameter  is  evidently 
twice  as  great  as  that  of  a  single  wire.  Two  wires  side 
by  side  have,  therefore,  but  half  as  much  resistance  as 
one  wire.  The  resistance  of  a  pure  metal  varies  greatly 
with  its  temperature,  increasing  as  the  temperature  in- 
creases. There  is  good  reason  to  believe  that  at  abso- 
lute zero  all  pure  metals  would  conduct  perfectly. 
Certain  alloys  have  nearly  the  same  resistance  for  a 
considerable  range  of  temperature.  'German  silver  is 


130  ELECTRICITY  AND  MAGNETISM. 

an  alloy  of  this  sort,  which  is  much  used  in  making 
standards  of  resistance. 

90.  Ohm's  Law.  Units.  —  We  have  learned  that 
a  current  flows  because  of  a  difference  of  potential,  and 
that  the  intensity  of  the  current  is  proportional  to  that 
difference  of  potential.  We  have  further  seen  that  the 
intensity  of  the  current  is  diminished  as  the  resistance 
of  the  conductor  increases.  Ohm  wrote  : 

*=f 

where  i  denotes  intensity  of  current  (or  current  simply), 
E  denotes  difference  of  potential  (sometimes  called 
electro-motive  force),  and  R  denotes  the  total  resistance 
of  the  circuit.  It  follows  that : 

(15a)    E—  Ri  and  (756)    R  =  — 

*  i 

The  units  in  everyday  use  for  measuring  these  quan- 
tities are  named  for  three  men,  who  laid  the  foundation 
for  our  knowledge  of  electric  currents.  Intensity  of 
current,  i,  is  measured  in  amperes,  difference  of  poten- 
tial, E,  is  measured  in  volts,  while  resistance,  R,  is 
measured  in  ohms : 

volts 

amperes  = 

ohms 

An  ohm  is  the  resistance  of  a  thread  of  mercury  1  mm. 
in  cross  section  and  106  cm.  long.  A  volt  is  the  differ- 
ence of  potential  between  the  terminals  of  a  gravity  cell. 
An  ampere  is  the  current  which  would  flow  through  a 


USEFUL  HEAT  EFFECTS.  131 

resistance  of  one  ohm  with  a  difference  of  potential  of 
one  volt.  More  exact  definitions  than  these  may  be 
given,  but  these  will  answer  all  practical  purposes. 

We  shall  get  some  notion  of  the  magnitude  of  the 
units  by  employing  them  in  various  ways. 

91.  Useful  Heat  Effects.  — By  using  good  conduct- 
ors in  a  circuit,  except  at  particular  points  which  have 
a  high  resistance,  we  may  obtain  intense  heat  at  those 
points.  In  the  two  forms  of  electric  lighting  in  com- 
mon use  this  is  done.  The  arc  light  consists  of  two 
rods  of  carbon  the  ends  of  which  are  allowed  to  touch, 
closing  the  circuit.  The  resistance  at  the  point  of  con- 
tact (the  contact  being  poor)  is  high,  the  carbons  be- 
come heated,  and  are  then  drawn  a  short  distance  apart. 
The  current  continues  to  pass  the  gap,  following  the 
heated  carbon  vapor ;  a  small  hollow  or  crater  is  formed 
in  the  positive  carbon,  which  is  heated  white  hot  and 
furnishes  most  of  the  light.  The  positive  carbon  is 
usually  placed  uppermost,  so  that  the  light  from  the 
crater  may  be  thrown  downward  (see  Fig.  89).  The 
temperature  of  the  arc  is  estimated  at  4,000°  C.  It  is 
the  highest  temperature  which  can  be  obtained  artifi- 
cially, and  is  sufficient  to  volatilize  all  substances.  The 
light  is  by  far  the  most  intense  of  any  artificial  light 
known.  The  arc  lamp  requires  about  6  or  8  amperes 
at  50  volts  and  gives  1,000  candle  power.  Various 
mechanical  and  electrical  devices  are  used  to  open  the 
arc  after  the  current  has  started  and  to  feed  the  carbons 


132 


ELECTRICITY  AND  MAGNETISM. 


together  as  they  waste  away.  Arc  lamps  are  usually 
connected  in  series,  so  that  the  total  resistance  of  the 
line  is  the  sum  of  the  resistances  of  all 
the  lamps  (see  Fig.  90).  A  line  of 
twenty  lamps  would  have  a  resistance  of 
some  tiling  like  125  ohms  and  would  re- 
quire a  difference  of  potential  of  1,000 
volts.  As  the  negative  terminal  of  the 
dynamo  and  the  farther  end  of  the  line 
are  usually  connected  to  ground,  it  is  not 
safe  for  a  person  standing  011  the  ground 
or  on  a  wet  floor  to  touch  such  a  circuit. 
For  lighting  houses  the  incandescent 
lamp  is  used.  It  consists  of  a  thread  or 
filament  of  carbon,  often  a  carbonized 
silk  thread,  sealed  within  a  glass  bulb  and 
connected  with  the  circuit  by  platinum 
wires,  which  are  sealed  into  the  glass. 
The  air  is  exhausted  from  the  bulb  so  that  the  carbon, 
when  heated  white  hot  by  the  current,  is  not  burned. 


FIG.  89. 


FIG.  90. 


The  common  size  of  incandescent  lamp  gives  16  candle 
power  and  uses  0.5  ampdre  at  100  volts,  or  1  ampdre 


USEFUL  HEAT  EFFECTS.  133 

at  50  volts.  Incandescent  lamps  are  connected  parallel, 
the  two  line  wires  being  kept  at  a  constant  difference 
of  potential  (see  Fig.  91).  The  more  lamps  there  are 
connected  at  any  one  time,  the  more  current  flows, 
or,  in  other  words,  the  less  the  resistance  of  the  circuit. 
A  lamp  is  turned  on  by  closing  a  gap  in  the  circuit  by 
means  of  a  key.  Electric  lamps  give  off  no  smoke  and 
consume  no  oxygen  from  the  air.  They  must,  however, 
be  used  full  power  or  not  at  all,  as  commonly  made,  and 
are  more  expensive  than  the  so-called  incandescent  gas 
lamp  invented  by 

Welsbach   for    the  j>     <j> 

same     illuminating 
power.     The  ability 

to  turn  the  lamp  on    o— 

or  off  at  a  moment's 

notice  and  to  control 

a  number  of   lamps 

in  a  ceiling  or  other 

inaccessible  place  from  a  single  convenient  point  is  a 

great  convenience.     Moreover,  the  incandescent  lamp  is 

perfectly  cleanly  and  requires  no  attention  whatever. 

The  electric  heating  of  houses  and  cooking  by  the 
heat  from  the  current,  while  perfectly  possible  and  very 
convenient,  have  not  become  common  because  too  ex- 
pensive. 

The  welding  of  metals  by  the  electric  current  is 
accomplished  by  connecting  the  metals  in  the  circuit 
as  the  carbons  of  an  arc  lamp  are  connected.  The  heat 


134  ELECTRICITY  AND   MAGNETISM. 

generated  at  the  point  of  contact  increases  the  resistance 
at  that  point,  which  still  further  heats  the  metal  until 
it  is  soft  enough  to  unite,  when  the  ends  are  pushed  to- 
gether and  the  current  is  cut  off.  A  very  large  current 
of  low  potential  is  required  for  this  work. 

92.  The  Current  Produced  by  Heat.  —  The  contact 
difference  between  two  metals  in  air  is  different  for  dif- 
ferent temperatures.  If  two  strips  of  unlike  metals  are 
connected  to  form  a  closed  circuit,  there  will  be  no  cur- 
rent flowing  in  the  circuit  as  long  as  the  two  places  of 


FIG.  92. 

union  (^A,  B,  Fig.  92)  are  at  the  same  temperature,  since 
the  differences  of  potential  at  A  and  B  are  the  same : 
but  if  A  be  placed  near  a  flame  while  B  is  kept  cool,  the 
differences  of  potential  at  A  and  B  will  not  be  the  same, 
and  a  current  will  flow  through  the  circuit  so  long  as 
the  points  A  and  B  are  kept  at  different  temperatures. 
A  series  of  such  elements  is  shown  in  Fig.  93.  A  series 
of  50  or  100  such  elements  made  of  antimony  and 
bismuth  and  joined  in  series  with  a  delicate  galvanom- 
eter make  an  exceedingly  sensitive  thermometer. 

Thermo-electric  generators  are  now  made  which  take 


MOT10X  PRODUCED  BY   THE   CURRENT.        135 


the  place  of  batteries,  the  heat  from  a  single  Bunsen 
burner  furnishing  a  difference  of  potential  of  5  volts 
and  a  current  on  short  circuit  of  4  ampdres.  Such  a 
generator  serves  admirably  for  charging  storage  batteries 
and  for  electroplating 
and  the  running  of  small 
motors.  The  difference 
of  potential  is  very 
constant. 


FIG. 


93.  Motion  Produced 
by  the  Current.  —  We 
have  seen  that  a  mag- 
netic needle  is  deflected 
by  the  passage  of  a  cur- 
rent in  its  neighborhood  because  the  conductor  is  sur- 
rounded  by  a  field   of   magnetic   force.     Two    parallel 
currents  will  attract  or  repel  each  other,  depending  on 


FIG.  94. 

their  direction  with  reference  to  each  other.  If  going 
in  the  same  direction  they  will  attract,  if  going  in 
opposite  directions  they  will  repel.  The  lines  of  force 
always  tend  to  neutralize  each  other  just  as  electric 
charges  do,  or  as  the  two  opposite  stresses  at  the  two 


136 


ELECTRICITY  AND   MAGNETISM. 


ends  of  a  stretched  cord  do.  At  A  (Fig.  94),  where  the 
currents  are  both  flowing  into  the  paper,  the  magnetic 
lines  are  seen  to  be  opposite  and  the  currents  attract ; 
at  B,  where  the  currents  flow  in  opposite  directions,  the 
magnetic  lines  between  the  wires,  being  alike,  repel. 

94.  Solenoids.  Electro-magnets.  —  A  long  wire 
wound  spirally  about  a  cylinder  has,  when  carrying  a 
.current,  a  field  of  magnetic  force  about  it  which  is  iden- 
tical with  the  field  of  a  bar  magnet  (see  Fig.  95). 


FIG.  95. 

Two  such  solenoids  behave  toward  each  other  like  two 
bar  magnets.  If  a  core  of  soft  iron  is  placed  in  the 
solenoid  the  field  of  force  becomes  very  much  stronger ; 
indeed",  such  an  electro-magnet,  when  carrying  even  a 
moderately  strong  current,  is  more  powerful  than  a  good 
bar  magnet.  Its  chief  usefulness  depends  upon  the 
fact  that  its  strength  may  be  increased  or  diminished  at 
will  by  increasing  or  diminishing  the  current,  the  mag- 
netism disappearing  almost  wholly  the  instant  the 
circuit  is  broken.  Electro-magnets  are  oftenest  made  in 
the  horseshoe  form  to  give  them  great  lifting  power  or 


THE  ELECTEO-MAGNETIC   TELEGEAPH.         137 

to  make  the  path  of  the  lines  of  force  as  much  in  the 
iron  as  possible,  rather  than  in  air,  where  they  are  less 
intense. 

95.  The  Electro-magnetic  Telegraph.  —  An  electro- 
magnet having  a  piece  of  soft  iron  which  is  supported 
by  a  hinged  bar  and  held  away  from  the  magnet  by  a 
spring  constitutes  what  is  known  as  a  sounder.  The 
movable  piece  of  iron  is  called  the  armature.  The 
sounder  is  the  instrument  used  for  receiving  messages, 
the  sending  being  done  with  a  simple  switch  called  a 


FIG.  96. 

key.  Let  the  key,  K,  be  open,  the  spring  holds  the 
armature  up  against  the  stop,  s.  Closing  the  key 
allows  the  battery  current  to  flow  through  the  electro- 
magnet, the  armature  is  drawn  down,  strikes  the  stop,  V, 
and  makes  a  sound.  When  the  key  is  opened  the 
magnet  no  longer  acts  and  the  spring  draws  the  arma- 
ture back  against  s.  In  Morse's  printing  telegraph, 
which  is  now  but  little  used,  a  point  on  the  stop  made  a 
dot  on  a  moving  paper  if  the  circuit  was  kept  closed 
a  very  short  time,  or  a  dash  if  the  key  was  kept  closed 
a  longer  time.  Although  the  messages  are  now  read 


138 


ELECTRICITY  AND  MAGNETISM. 


almost  universally  by  ear,  the  signals  are  called  dots  and 
dashes,  and  various  combinations  of  dots  and  dashes 
constitute  the  alphabet.  The  code  of  signals  is  as 
follows : 


a  — 

1} 

c  -  -    - 

d 

e  - 

f 

9- 

h 

»--  - 

j 


k 


u  

1  

V  

2  

w  

3  

X  

4  

y--  -- 

5  

iy    _ 

« 

6   ™ 

&-    --- 

7- 

i 

8  

f  

9  

n 

It  is  evident  that  by  closing  K  a  signal  will  be  made 
at  the  sounder,  no  matter  how  far  away  K  is  placed, 
provided  that  the  battery  is  strong  enough  to  send  a 
current  of  sufficient  strength  through  the  line.  The 
resistance  of  a  long  telegraph  line  is  several  thousand 
ohms,  and  the  currents  are  weak  even  when  the  return 
circuit  is  made  through  the  earth. 

For  operating  on  long  lines  a  device  called  a  relay  is 
used.  It  consists  of  an  electro-magnet  having  a  large 
number  of  turns,  small  wire  being  used  so  as  to  bring  the 
current  close  to  the  iron  core.  The  armature  of  a  relay 
is  hinged  to  move  with  a  small  force  so  that  a  weak  cur- 
rent can  actuate  it.  This  armature  when  drawn  toward 
the  magnet  acts  as  a  key  to  close  a  local  circuit,  having 
a  battery  and  sounder.  The  local  circuit  has  a  low  re- 
sistance, and  is  readily  operated  by  a  few  cells  with  force 


ELECTRIC  BELLS. 


139 


enough  to  give  as  loud  a  sound  as  is  desired.     A  line 
containing  a  relay  and  local  circuit  is  shown  in  Fig.  97. 


FIG.  97. 

All  of  the  instruments  are  duplicated  at  every  office  on 
the  line.     Every  key  has  a  switch,  which  is  kept  closed 
except  when  the  operator  is  sending  a  mes- 
sage.    Each  office  has  a  particular  letter  by 
which  it  is  called.     Every  office  on  the  line 
may  read  any  message  which  goes  over  the 
line.     When  a  particular  office  is  wanted 
its  call  is  repeated  till  the  operator  answers 
by  opening  his  switch,  after  which  he  closes 
the  switch  and  receives  the  message. 

96.  Electric  Bells.  —  The  electric  bell 
is  arranged  to  break  its  own  circuit  as  fast 
as  it  is  closed,  thus  keeping  up  a  succession 
of  taps  as  long  as  the  key  or  push  button 
is  kept  closed.     Its  connections  are  shown 
in  Fig.  98.     When  the  circuit  is  closed  by          FlG*  98> 
pushing  the  button  at  j9,  the  armature,  «,  is  drawn  toward 
the  magnet  and  the  hammer  strikes  the  bell.     By  mov- 


140 


ELECTRICITY  AND   MAGNETISM. 


ing  away  from  the  stop,  «,  the  armature  breaks  the  cir- 
cuit, the  spring  draws  the  armature  back,  and  the  cycle 
of  operations  is  repeated. 

97.  Electric  Motors.  — An  electric  motor  consists  of 
an  electro-magnet  in  the  field  of  which,  another  electro 
magnet  is  made  to  revolve  continuously  by  making  and 


FIG.  99. 


breaking  the  circuit  through  it.  The  stationary  magnet 
is  called  the  field  magnet,  the  movable  one  the  arma- 
ture. It  is  evident  that  if  a  current  is  made  to  flow 
through  ns  (see  Fig.  99)  when  it  is  in  the  vertical 
position,  it  will  rotate  in  the  direction  of  the  arrow. 
Now  if,  when  ns  has  reached  the  horizontal  position,  the 
current  in  it  be  reversed,  thus  reversing  the  poles  in 


ELECTRIC  MOTORS. 


141 


the  armature,  the  latter  will  continue  to  rotate  in  the 
same  direction.  The  armature  is  mounted  on  a  shaft 
which  rotates  between  the  poles  of  the  field  magnet 
and  the  reversal  of  the  current  is  accomplished  by 
means  of  a  device  called  a  commutator  (see  Fig.  100). 
The  commutator  consists  of  a  divided  ring  fastened  to 
the  shaft  and  insulated 
from  it.  The  ends  of 
the  wire  which  form  the 
coils  of  the  armature  are 
connected  to  the  two 
halves  or  segments  of 
the  commutator.  Two  "  FIG.  100. 

strips  of  copper  or  carbon  called  brushes  are  supported 
in  a  fixed  position  so  as  to  rub  upon  the  rotating  com- 
mutator. The  wires  carrying  the  circuit  are  attached 
to  these  brushes.  It  is  evident  that  as  the  armature 
rotates,  the  direction  of  the  current  in  the  armature 

changes  once  during 
each  revolution.  The 
motion  will  be  more 
steady  if  the  armature 
have  two  coils  at  right 
angles  to  each  other. 
The  commutator  must 
then  have  four  segments  and  each  coil  will  be  idle  half 
the  time  (see  Fig.  101).  In  large  motors  the  commu- 
tator often  has  120  or  more  segments.  A  typical  motor 
is  shown  in  Fig.  102. 


FIG>101- 


142 


ELECTRICITY  AND  MAGNETISM. 


98.  Currents  Produced  by  Motion.  —  We  have  seen 
that  currents  may  be  the  result  of  chemical  action  and 
that  chemical  action  may  be  produced  by  a  current, 
likewise  that  currents  may  be  produced  by  heat  and 
that  heat  is  an  effect  easily  produced  by  the  current. 
It  is  perhaps  to  be  expected,  then,  that  since  motion  may 
be  produced  by  a  current,  a  current  might  be  produced 
by  motion.  Such  is,  indeed,  the  case  whenever  a  con- 
ductor is  moved 
across  the  lines  of 
magnetic  force. 
This  fact,  which 
was  discovered  by 
Faraday  in  1837, 
is  of  the  greatest 
importance.  Such 
currents  are  known 
as  induced  cur- 
rents. The  laws 
which  govern  their 
production  may  easily  be  illustrated  by  means  of  a  coil 
of  wire  attached  to  a  galvanoscope,  a  bar  magnet,  and 
an  electro-magnet  having  a  removable  core.  The  coil 
attached  to  the  galvanoscope  is  usually  referred  to  as  the 
secondary  circuit  (see  Fig.  103).  The  electro-magnet 
is  called  the  primary  circuit.  The  galvanoscope  should 
be  far  enough  removed  so  that  its  needle  will  not  be 
much  disturbed  by  the  movements  of  the  magnet.  If 
we  thrust  a  bar  magnet  into  the  coil  the  needle  of 


FIG.  102. 


CURRENTS  PEODUCED  BY  MOTION. 


143 


the  galvanoscope  will  be  deflected,  but  when  the  mag- 
net comes  to  rest  the  needle  will  return  to  zero.  Now 
if  we  quickly  pull  the  magnet  out,  the  needle  will  be 
deflected  in  the  opposite  direction.  If  the  other  pole 
of  the  magnet  be  used 
the  deflection  will  be 
opposite  in  direction. 
The  primary  coil,  which 
is  a  magnet  when  a 
current  flows  through 
it,  will  produce  exactly 
the  same  effects,  the 
deflections  being  great- 
er when  the  core  is  in 
the  coil.  If  the  pri- 
mary coil  be  placed 
in  the  secondary  coil 
while  connected  with 
the  battery  and  a  vari- 
able resistance,  cur- 
rents will  be  induced 
if  we  suddenly  increase  or  diminish  the  resistance. 
Similarly,  if  we  break  or  make  the  primary  circuit,  cur- 
rents will  be  induced  in  the  secondary. 

The  quickness  with  which  the  change  is  made  in  all 
the  cases  mentioned  determines  the  difference  of  poten- 
tial produced.  It  will  thus  appear  that  we  are  not 
limited  as  we  are  in  the  case  of  batteries  and  thermo- 
piles, but  may  produce  very  high  potentials  without 


FIG.  103. 


144 


ELECTRICITY  AND  MAGNETISM. 


very  high  resistances.  All  that  is  required  is  that  there 
shall  be  a  conductor  moving  rapidly  in  a  field  of  mag- 
netic force,  or,  better,  a  field  of  force  rapidly  changing 
in  intensity  in  the  neighborhood  of  the  conductor. 

99.  Spark  Coils. — A  primary  circuit  having  an 
interrupter  like  that  of  an  electric  bell  will,  if  it  con- 
sist of  a  large  number  of  coils,  induce  at  the  instant 
the  circuit  is  broken  a  difference  of  potential  in  itself 
which  opposes  the  breaking  of  the  circuit  and  produces 

a  spark.  In 
this  case  each 
coil  of  the  cir- 
cuit acts  upon 
its  neighbors 
PI  ^u  and  the  pro- 

cess is  called 
s  e  1  f -indue  tion. 
Such  simple  spark  coils  are  used  for  lighting  gas  and  to 
a  limited  extent  for  medical  purposes.  The  common 
form  of  induction  coil  (Ruhmkorff's,  Fig.  104)  has  but 
few  turns  in  the  primary  coil,  while  the  secondary, 
which  is  around  it,  has  a  very  large  number  of  coils  oi 
fine  wire.  Such  coils  are  used  to  illuminate  vacuum 
tubes  and  for  medical  purposes.  The  terminals  of  the 
primary  are  usually  connected  to  a  condenser  which 
reduces  the  self-induced  spark  in  the  primary.  This 
primary  spark  is  a  disadvantage  in  two  ways :  it  burns 
the  contacts  and  it  diminishes  the  spark  in  the  secondary 


.  104. 


THE  DYNAMO. 


145 


by  retarding  the  break  in  the  primary.  The  spark  pro- 
duced by  an  induction  coil  is  identical  in  its  effect 
with  that  produced  by  the  Toepler-Holtz  machine. 

100.  The  Dynamo.  — Any  electric  motor  if  connected 
to  a  galvanometer  and  turned  by  hand  will  show  a  current 
through  the  galvanometer.  Indeed,  the  conditions  for 
generating  a  strong  continuous  current  are  exactly  met 
in  the  motor. 

Suchama-  i  (•) 

chine    when  «       A* 

used  for  pro- 
ducing a  cur- 
rent is  called 
a  dynamo-^ 
electric  ma- 
chine or,  more 
briefly,  a  dy- 
namo. The 
construction 
of  the  m a- 

chine  we  need  not  describe  again.  It  is  identical  with 
the  motor.  The  field  magnets  have  a  small  amount  of 
permanent  magnetism  —  sufficient  to  start  a  small  cur- 
rent in  the  armature  coil  as  soon  as  it  begins  to  rotate. 
This  current,  or  part  of  it,  flows  through  the  coils  of 
the  field  magnet  and  the  potential  soon  rises  to  a  maxi- 
mum and  remains  steady  for  a  constant  speed  and  load. 
In  order  to  explain  the  induced  current  in  terms  of  the 


FIG.  105. 


146 


ELEOTEICITY  AND  MAGNETISM. 


magnetic  field  we  must  now  recall  that  a  magnetic  field 
is  thought  of  as  filled  with  elastic  lines  of  force  which 
tend  to  contract  lengthwise  and  mutually  repel.  A 
wire  carrying  a  current  has  lines  of  force  about  it.  If 
these  lines  of  force  about  the  wire  are  more  numerous 
in  one  part  of  a  circuit  than  in  another  their  mutual 

repulsion  causes 
them  to  spread 
out  and  equalize 
the  potential. 
How  does  a  wire 
which  moves 
across  lines  of 
force  get  the 
lines  wrapped 
about  it  —  that 
is,  get  a  current 
flowing  through 
it  ?  Let  the  row 
of  black  dots  in 
Fig.  105  repre- 
sent the  succes- 
sive positions  of 
a  wire  which  is  cutting  through  a  line  of  force  as  it 
moves  downward  through  the  field.  The  line  stretches 
more  and  more  till  it  finally  breaks,  reuniting  above 
and  leaving  a  line  of  force  about  the  wire  in  the  direc- 
tion corresponding  to  the  lines  about  a  current  which  is 
flowing  into  the  paper.  At  B  a  wire  moving  upward 


OCOOOOG 


D 
FIG.  106. 


VERSITY 


THE  DYNAMO. 


147 


would  have  lines  about  it  corresponding  to  a  current 
out  of  the  paper.  If  A  and  B  are  the  upper  and  lower 
sides  of  a  coil  then  the  current  flows  round  the  coil. 
The  lines  (rings)  of  force  which  are  constantly  forming 
at  A  and  B  are  as  constantly  spreading  out  along  the 
wire  to  equalize  the  potential.  See  a  top  view  in  Fig. 
106,  where  it  may  be  seen  that  the  lines  of  force  which 


X^^'\'^^^\y*\/^^v^^''\^*r/l^'>''C'^ 

/  /  V  v  v  y  \.  x  <  y  y\  y  X  *  *  X  X  /  X  X  <  \ 

I /  XX  KM  A  X  XX  XX  ^  XX  XX  XX  X^ 


FIG.  107. 

reach  0  and  D  from  the  two  points  A  and  B  are  going 
in  the  same  direction  and  hence  the  current  flows  in 
one  direction  in  all  parts  of  the  wire.  When  A  is  at 
the  top  and  B  is  at  the  bottom  they  are  moving  parallel 
to  the  lines  of  force  and  cut  none  of  them.  They 
therefore  generate  no  current  at  this  part  of  the  revolu- 
tion, while  a  maximum  current  is  generated  when  the 
coil  is  cutting  squarely  across  the  lines.  When  A  be- 


148 


ELECTRICITY  AND  MAGNETISM. 


gins  to  ascend  and  B  to  descend  the  current  will  be 
reversed  in  the  coil,  but  owing  to  the  commutator  it 
will  always  flow  in  one  direction  through  the  external 
circuit  at  D.  It  will  be  seen  that  a  dynamo  having  but 
one  coil  and  but  one  pair  of  segments  in  its  commuta- 
tor would  give  a  very  unsteady  current.  Increasing 
the  number  of  coils  increases  the  steadiness  of  the 

current.  In  Fig. 
107  the  current 
strength  for  suc- 
cessive short  inter- 
vals during  three 
revolutions  of  the 


FIG.  108. 


dynamo  is  shown 
by  the  curves.  At 
A  the  current  falls 
to  zero  twice  dur- 
ing each  revolu- 
tion. At  B  with 
two  coils  it  falls  but  .7  as  low,  at  0  with  4  coils  but 
.9  as  low,  and  at  Z>,  where  24  coils  are  used,  the  current 
does  not  vary  to  any  perceptible  extent.  A  modern 
dynamo  is  shown  in  Fig.  108. 

101.  Alternators.  —  For  certain  purposes  an  alternat- 
ing current  has  advantages  over  a  direct  current.  If  we 
solder  the  ends  of  our  armature  coil  to  a  pair  of  rings 
which  are  not  split,  but  placed  side  by  side  on  the  shaft 
and  insulated  from  the  shaft  and  from  each  other,  the 


AL  TERN  A  TORS.  149 

brushes  which  rest  on  these  rings  will  convey  to  the 
circuit  the  identical  current  which  flows  in  the  arma- 
ture coil,  that  is,  an  alternating  current.  It  is  desirable 
that  the  alternations  occur  very  frequently.  The  field 
magnet  is  therefore  usually  made  with  a  number  of 
poles  wound  so  that  alternate  poles  are  north  poles. 
The  coil  then  passes  from  a  north  to  a  south  pole  sev- 


FlG.  109. 


eral  times  during  each  revolution.  The  field  magnets 
cannot  be  maintained  by  the  alternating  current,  as  they 
would  be  continually  changing  polarity.  It  is  usual  to 
employ  the  current  from  a  small  direct  current  dynamo 
for  the  purpose.  Fig.  109  shows  an  alternating  current 
dynamo  with  the  accompanying  direct  current  exciter 
driven  by  a  belt  from  the  shaft  of  the  alternator. 


150 


ELECTRICITY  AND  MAGNETISM. 


102.  The  Transformer. — If  the  current  from  an 
alternator  is  sent  through  the  primary  coil  of  an  induc- 
tion coil  while  the  hammer  is  wedged  shut  fast,  an 
induced  alternating  current  of  much  higher  potential 
but  smaller  in  amount  will  flow  through  the  secondary. 
If  the  current  from  the  alternator  is  sent  through  the 
long  secondary  coil  a  large  current  of  low  potential 
will  flow  in  the  short  coil  of  the  primary.  It  is  very 

desirable  i  n 
the  distribu- 
tion of  elec- 
tric currents 
over  large 
towns  to  use 
high  poten- 
tials, since 
this  makes 
possible  the 
use  of  small 
wires  for  con- 
veying  the 

currents.  These  high  potential  currents  are  not  safe  for 
common  people  to  handle,  but  by  sending  them  through 
a  transformer  they  may  be  changed  to  any  desired  poten- 
tial. The  transformer  consists  of  an  iron  core  around 
one  part  of  which  are  wound  a  large  number  of  turns  of 
fine  wire,  which  carries  the  high  potential  current,  P, 
from  the  dynamo  (Fig.  110).  Around  the  other  part  are 
wound  a  few  turns  of  large  wire,  $,  which  carry  a  low 


EXEECISES.  151 

potential  current  to  the  house.  Such  currents  are 
exactly  as  good  for  use  in  incandescent  lamps  as  direct 
currents ;  for,  although  the  current  falls  to  zero  at  every 
alternation,  the  alternations  occur  so  often  (more  than 
a  hundred  times  per  second)  that  the  filament  of  the 
lamp  has  no  time  to  fall  in  temperature  between  times. 

Exercises. 

70.  Connect  the  zinc  and  copper  plates  of  a  gravity  cell  to  a 
galvanoscope,  but  use  water  with  a  few  c.c.  of  sulphuric  acid 
instead  of  the  regular  copper  sulphate  solution.     The  instru- 
ment must  be  sensitive  enough  to  deflect  at  least  twenty  degrees. 
Note  the  changes  in  deflection  for  half  an  hour.     Now  remove 
about  half  of  the  acidulated  water  and  replace  it  with  copper 
sulphate  solution  and  add  a  small  handful  of  copper  sulphate 
crystals.     Note  the   deflections  for  half   an   hour.     Test  the 
same  battery  with  the  same  instrument  every  day  for  a  week, 
keeping  the  circuit  closed  between  times  through  some  resist- 
ance like  a  telegraph  sounder. 

71.  (a)  Connect  through  a  galvanoscope  or  an  ammeter  each 
of  two  gravity  cells  separately,  then  the  two  in  series,  then  the 
two  parallel.     (6)  Compute  what  current  you  ought  to  get  in 
each  case  if  the  cells  had  each  a  potential  difference  of  1  volt 
and  a  resistance  of  5  ohms.     What  is  the  sum  of  the  potential 
differences  in  each  case?  of  the  resistances  ? 

72.  What  is  the  effect  on  the  potential  difference  of   a   cell 
of :  (a)  doubling  the  size  of  the  plates,  (6)  moving  the  plates 
nearer  together,  (c)  effect  on  resistance  of  doubling  the  size, 
(d)  effect  on  resistance  of  bringing  the  plates  nearer  together  ? 

73.  Six  gravity  cells,  each  having  a  potential  difference  of 
1  volt  and  a  resistance  of  5  ohms,  may  be  connected  in  four 
different  ways  as  shown  in  Fig.  111.     Which  is  the  best  way 
to  connect  them  when  the  largest  obtainable  current  is  desired  : 


152 


ELECTRICITY  AND   MAGNETISM. 


(a)  through  a  low  resistance,  say  1  ohm,  (6)  through  7  ohms, 
(c)  through  40  ohms  ? 

74.  What  sort  of  battery  would   best   be   selected    (a)   for 
electric  bell  work,  (6)  for  an  induction  coil,  (c)  for  telegraphy  ? 

75.  Two  metals,  bismuth  and  antimony,  have  a  contact  dif- 
ference of  potential  which  is  changed  .000117  volt  for  1°  C. 
How   many   bismuth-antimony   pairs   must   be    connected   in 
series  to  give  a  potential  difference  of  1  volt  if  the  tempera- 
ture difference  is 
100°  C.  ? 

76.  A  dynamo 
giving  a  constant 
difference  of 
potential  of  55 
volts  is  con- 
nected to  a  cir- 
cuit wired  for  20 
lamps,  any  one 
of  which  may  be 
turned  on  or  off 
at  pleasure. 
Each  lamp  has 
a  resistance, 
when  hot,  of  50 
ohms.  An  am- 
meter is  connected  in  the  circuit.  What  current  will  the 
ammeter  record  when  one  lamp  is  turned  on  ?  10  lamps  ? 

77.  A  dynamo  intended  to  be  used  with   55  volt  lamps   is 
found  to  give  but  48  volts  when  run  at  1,600  revolutions  per 
minute.     How   many  revolutions  should  it  make   to   give   55 
volts  ? 

78.  Beneath  a  large  shallow  flat  dish  (see  Fig.  112)  place  a 
sheet  of  cross-section  paper  having  a  number  in  each  square. 


FIG.  in. 


EXEECISES. 


153 


Put  enough  acidulated  water  in  the  dish  to  fill  it  to  a  depth  of 
about  1  cm.,  and  dip  the  wires  from  a  battery  into  the  liquid, 
one  near  each  end  of  the  dish.  Now  place  one  wire  from  a 
galvanometer  at  any  point,  a,  near  one  side  of  the  dish  and 
find  a  point,  &,  on  the  other  side,  such  that  when  the  other 
wire  from  the  galvanometer  is  dipped  in  the  liquid  at  the  point 
no  current  will  flow.  Points  a  and  &  are  at  the  same  potential. 
Keeping  a  at  the  same  place  find  three  or  four  other  points 


FIG.  112. 

between  a  and  6  which  are  also  at  the  same  potential.  Indi- 
cate all  these  points  on  a  sheet  exactly  like  the  one  under  the 
dish  and  connect  them  by  a  line.  Plot  a  number  of  such  equi- 
potential  lines  and  then  draw  lines  from  Z  to  0,  crossing  the 
equipotential  lines  at  right  angles.  These  latter  are  lines  of 
force. 

79.  The  fall  of  potential  in  any  part  of  a  circuit  is  propor- 
tional to  the  resistance  of  that  part  of  the  circuit.  The  differ- 
ence of  potential  at  the  terminals  of  a  dynamo  is  found  by  the 
voltmeter  to  be  110  volts.  The  line  is  of  copper  wire,  No. 


154  ELECTBICITY  AND    MAGNETISM. 

12,  which  has  a  resistance  of  0.005  ohm  per  metre.  At  the 
end  of  the  line  1  km.  from  the  station,  what  is  the  difference 
of  potential  between  the  wires  ?  If  you  have  lamps  designed 
for  110,  105,  and  100  volts  which  should  you  use  at  the  end  of 
the  line?  Which  at  a  point  halfway?  Would  it  make  a  dif- 
ference whether  2,  10,  or  20  lamps  were  used  ?  Note  that  if 
the  potential  is  too  high  for  the  lamps  they  will  be  quickly 
"  burned  out,"  while  if  too  low  they  will  not  be  bright,  but  red. 

80.  (a)  Connect  a  long  wire  to  a  gravity  cell  and  determine 
the  direction  of  the  current  in  the  wire  by  means  of  a  compass 
needle.     (6)   Determine  in  the  same  way  the  direction   of   a 
current  from  a  storage  cell  or  two  cells  of  plung.e  battery,  then 
cut  the  wire,  bare  the  ends,  and  dip  them  into  acidulated  water. 
The  positive  wire  should  be  coated  with  copper  oxide  formed 
by  the  union  of  the  oxygen,  which  collects   there,  with   the 
copper. 

81.  A  current  of  1  ampere  deposits  0.003277  gram  of  copper 
per  second  from  a  solution  of  copper  sulphate.     What  is  the 
strength  of  a  current  which  deposited  1.26  grams  of  copper  in 
one  hour  ? 

82.  With  a  current  of  10  amperes  how  long  would  it  take  to 
deposit  a  layer  of  copper  .2  mm.  thick  over  an  electrotype  hav- 
ing a  surface  of  20  sq.  cm.  ? 

83.  Measure  the  length  and  diameter  of  a  long  piece  of  iron 
wire  such  as  is  used  for  supporting  stove  pipes  and  compute  its 
resistance,  taking  the  specific  resistance  of  iron  wire  at  20°  C. 
as  0.000014.     Connect  a  battery  through  the  iron  wire  and  an 
ammeter  and  jcompute  the  resistance  of  the  battery  and  am- 
meter combined.     The  resistance  of  the  ammeter  is  usually  so 
small  as  to  be  negligible.     Cut  the  wire  in  halves  and  try  the 
experiment   with   each   half   separately.     Compare   the   three 
values  obtained  for  the  resistance  of  the  battery.     It  is  not  to 
be  expected  that  they  will  agree  very  closely,  especially  if  the 
temperature  of  the  room  changes  much. 


EXERCISES.  155 

84.  Connect  a  galvanometer  to  a  Daniell  cell  and  hold  a  bar 
magnet  in  different  positions  above  the  needle  to  find  what 
positions  will  increase 
the  deflection  and  what 
positions  will  diminish 
it.  It  is  usually  possible 
so  to  adjust  the  field  that 
the  needle  may  be  made 
to  deflect  between  30° 
and  60°,  which  is  the 
best  amount  for  all  pur- 


poses.    It  is  useless  to    ^ 

try  to  measure  the  cur- 

i  a     *•         £  FlG-  113- 

rent  by  the  deflection  of 

a  galvanometer  when  the  deflection  is  nearly  90°  or  when  the 
deflection  is  exceedingly  small. 

85.  Suppose  two  small  needles  fastened  together  parallel  on 
one  support  with  their  north  poles  opposite,  as  shown  in  Fig. 
113.     Would  they  be  more  or  less  sensitive  to  the  effects  of  a 
current  through  the  coil  of  wire  than  one  of  them  alone  would 
be?     Such  an  arrangement  is  called  an   astatic   pair.     What 
does  astatic  mean? 

86.  Why  should  the  coils  of  a  galvanometer  always  be  placed 
north  and  south  when  the  galvanometer  is  to  be  used  ? 

87.  Connect  by  wires  two  metal  handles  to  the  terminals  of 
the  secondary  of  a  small  induction  coil  and  test  the  physiologi- 
cal effects  of  the  current.     Connect  the  same  handles  to  points 
on  opposite  sides  of  the  break  of  a  vibrating  electric  bell  in  such 
a  way  as  to  include  the  magnet  in  the  circuit  and  test  the  bell 
for   physiological  effects.     Has   the  bell  a  secondary  circuit? 
Whence  the  effect  ?     Test  in  the  same  way  the  strongest  bat- 
tery in  the  laboratory. 

88.  A  common  telephone  consists  of  a  bar  magnet  in  front 
of  which  is  supported,  in  the  rubber  enclosing  case,  a  sheet  of 


156 


ELECTRICITY  AND   MAGNETISM. 


soft  iron.  A  coil  of  wire  surrounds  the  magnet  near  the  iron 
disk.  When  the  coils  of  two  such  telephones  are  connected  in 
series  by  wires  as  in  Fig.  114,  words  spoken  at  T  may  be  heard 
at  T',  a  mile  or  more  away. 

(a)  Explain  how  the  motion  of  the  iron  diaphragm  at  T  pro- 


FIG.  114. 


duced  by  the  vibrations  of  the  air  causes  induced  currents  in 
the  coil,  which,  when  conducted  along  the  wire  to  T',  set  the 
diaphragm  of  that  telephone  yibrating.  (6)  Which  telephone 
acts  like  a  dynamo  and  which  like  a  motor?  Explain. 


CHAPTER  V. 
WORK.     ENERGY.     MACHINES. 

103.  Work.  Energy.  —  In  our  study  of  motions 
hitherto  we  have  seen  that  motion  of  any  sort  could 
produce  motions  of  other  sorts,  but  we  have  not  given 
much  attention  to  the  quantity  of  the  effect  which  any 
given  cause  produces.  We  have  studied  the  how  of 
the  phenomena  rather  than  the  how  much.  We  propose 
now  to  review  what  we  have  learned  and  see  if  the 
separate  topics  cannot  all  be  brought  into  closer  rela- 
tions than  we  have  yet  perceived  to  subsist  between 
them. 

We  shall  use  the  words  work  and  energy  in  a  some- 
what technical  sense  and  we  must  bear  in  mind 
that  they  are  so  used.  We  must  be  especially  careful 
to  note  the  distinction  between  force  and  work.  A  force 
is  measured  by  the  velocity  which  it  imparts  to  unit 
mass  in  unit  time.  Unit  force,  when  no  other  force  is 
acting,  gives  unit  velocity  to  unit  mass  in  unit  time.  A 
dyne  of  force  gives  to  one  gram  a  velocity  of  one  cen- 
timetre per  second  in  a  second  of  time.  If  this  unit 
force  moved  the  unit  mass  through  unit  distance  it  did 
unit  work  upon  it.  Work  is  measured  by  the  product 
of  the  force  acting  times  the  distance  through  which  it 
moved  the  body. 

157 


158  WORK.     ENERGY.     MACHINES. 

Work  =  force  X  distance 

s-in^  ^7  VV 

(16)     w—fl 

If  two  forces  balance  each  other  so  that  no  motion 
results  no  work  is  done. 

Any  units  of  force  and  distance  may  be  used  for 
measuring  work.  The  foot-pound  is  the  work  done  in 
moving  a  pound  one  foot  against  gravity.  The  erg  *  is 
the  work  done  when  the  force  of  a  dyne  produces 
motion  through  a  distance  of  J.  cm.  The  work  done 
in  lifting  a  gram  1  cm.  is  980  4\$K{m.  The  foot-pound 
is  equal  to  980  X  453.6  X  30.48  =  13,548,703  ergs. 
The  erg  is  well  adapted  to  the  measurement  of  small 
amounts  of  work,  the  foot-pound,  kilogram-metre,  and 
so  forth,  for  measuring  large  amounts. 

Any  body  which  has  motion  imparted  to  it  gains 
thereby  the  ability  to  put  other  bodies  in  motion,  or,  in 
other  words,  the  ability  to  do  work.  A  ball  fired  from 
a  gun  in  empty  space  would  do  no  work  until  it  should 
strike  some  body  which  offered  resistance  to  its  motion. 
When  it  had  done  an  amount  of  work  equal  to  that 
which  was  imparted  to  it,  it  would  come  to  rest.  Our 
earth  is  rushing  through  space  at  the  rate  of  18.5 
miles  per  second.  If  it  should  strike  another  planet 
it  would  do  work  on  a  magnificent  scale,  but  it  meets 
with  no  perceptible  resistance,  and  therefore  does  no 
work.  It  takes  work  to  lift  the  hammer  of  a  pile- 
driver  high  in  air.  So  long  as  the  hammer  hangs  there 

*  Erg  from  Greek  ergon,  work.    The  same  root  is  found  In  en-erg-y,  from 
Greek  energia. 


WORK.     ENERGY. 


159 


(see  Fig.  115)  it  possesses  the  ability  to  do  work  — 
loose  it  and  let  it  fall,  it  does  work.  The  ability  to  do 
work  we  call  energy.  The  energy  of  a  body  is  the 
measure  of  the  work  which  has  been  done  upon  it,  less 
what  it  has  since  done,  or  it  is  the  measure  of  the  work 
the  body  must  do  before  it  can  come  to  rest,  if  it  is  in 


FIG.  115. 

motion,  or  the  work  it  must  do  to  reach  a  condition  of 
perfectly  stable  equilibrium,  if  it  is  at  rest.  The  mov- 
ing bullet  and  the  lifted  hammer  both  possess  energy. 
Though  they  are  not  at  the  instant  doing  work,  it  took 
work  to  put  them  in  their  present  relations  to  other 
objects,  The  energy  of  the  bullet  is  energy  of  motion, 


160        WORK.  ENEEGT.  MACHINES. 

usually  called  kinetic  *  energy ;  the  energy  of  the  lifted 
hammer  is  energy  of  position  or  potential  energy. 
Kinetic  energy  always  implies  that  the  body  has  been 
acted  upon  by  a  force,  but  if  its  quantity  of  kinetic 
energy  is  not  changing  there  are  no  forces  now  acting. 
Potential  energy  implies  that  the  body  is  in  equilibrium 
under  a  balance  of  forces,  the  cessation  of  any  one  of 
which  would  result  in  motion.  A  bent  bow  is  held  in 
place  by  two  hands  pulling  apart,  the  hand  holding  the 
string  lets  go  and  the  arrow  speeds  away  with  energy, 
now  kinetic,  which  was  a  moment  ago  potential  in  the 
strained  bow  and  string.  The  sum  of  the  energy  of 
the  system  of  bodies  (bow  and  arrow)  is  unchanged. 
We  say  the  energy  has  been  transformed  in  kind,  but 
unchanged  in  amount. 

104.  Conservation  of  Energy.  —  No  physical  change 
can  occur  without  a  transfer  of  energy,  and  conversely, 
every  transfer  of  energy  involves  some  sort  of  change 
in  the  motions  of  at  least  two  bodies.  We  shall  call 
all  the  bodies  concerned  in  any  particular  transfer  of 
energy  a  system.  In  all  experiments  where  careful 
measurements  are  possible,  it  is  found  that  similar 
measured  quantities  of  energy  will  always  produce 
equal  effects  of  any  particular  kind.  A  moving  ball 
which  has  a  certain  mass  and  volocity  will  generate  a 
perfectly  definite  amount  of  heat  when  it  strikes.  A 
certain  strength  of  electric  current  will  produce  in  a 

*  Greek,  kineo,  move. 


CONSERVATION   OF  ENERGY.  161 

given  time  a  perfectly  definite  amount  of  heat,  or  chem- 
ical action,  or  mechanical  motion.  The  amount  of 
chemical  action,  or  heat,  or  mechanical  motion,  which 
was  produced  by  the  current  mentioned,  would,  if 
spent  entirely  in  producing  current,  produce  exactly 
the  quantity  of  electrical  energy  with  which  we  started. 

The  most  serious  practical  difficulty  met  with  in 
carrying  out  such  experiments  is  the  tendency  of  energy 
to  distribute  itself  to  neighboring  bodies  in  the  form  of 
heat.  All  careful  experiments,  however,  lead  to  the 
one  conclusion,  —  that  energy  is  never  gained  or  lost  in 
any  transfer,  or,  in  other  words,  the  quantity  of  energy 
of  a  system  of  bodies  concerned  in  a  transfer  of  energy  is 
absolutely  constant.  This  law,  which  is  the  product  of  no 
one  man's  labor,  but  represents  the  sum  of  the  results 
of  thousands  of  students  of  natural  phenomena,  is  the 
great  fundamental  law  of  physics.  It  is  known  as 
the  Law  of  Conservation  of  Energy. 

If  w  be  the  quantity  of  energy  lost  by  any  one  of 
a  system  of  bodies  and  w',  w",  etc.,  be  the  quantities 
gamed  by  the  other  bodies  in  the  transfer,  then  : 
(17)     w  =  wf  +  w"  +  etc. 

If  e,  d ,  e",  etc.,  represent  the  energy  of  each  body  of 
the  system  at  any  instant,  then: 

(18)     e  +  e'+e"  +  etc.  =  constant 
so  long  as  the  system  is  kept  from  contact  with  bodies 
outside  the  system.     In  practice  such  perfect  isolation 
of  a  system  of  bodies  is  exceedingly  difficult  to  secure. 


162 


WOEK.     ENERGY.     MACHINES. 


The  more  nearly  perfect  the  isolation,  however,  and  the 
more  careful  the  measurements,  the  more  nearly  the  re- 
sults will  be  found  to  agree  with  the  law.  It  is  evident 
that  we  require  for  a  systematic  study  of  energy  a 
system  of  units  which  shall  comprise  a  separate  unit 
for  each  sort  of  energy  and  a  system  of  equivalents 
by  which  energy  measured  in  one  form  may  be  com- 
pared with  that  measured  in  other  forms. 

105.  Units  of  Energy.     Equivalents.  —  The  poten- 
tial   energy  of    a   stretched  spiral   spring,   as   a  spring 
balance,  is  measured  directly  in  ergs 
if  we  know  what  force  was  used 
in  stretching  the  spring  and  how 
much  its  length  was  increased,  or, 
more  exactly,  how  much  the  tlis* 
^       E>   ,CO       tance  between  the  points  of  appli- 
6  cation,  A,  B  (Fig.   116),    of   the 

B  two  opposing  forces  was  changed 
under  the  influence  of  the  force. 
The  potential  energy  of  a  ball 
which  has  been  lifted  to  a  height 
I  is  measured  in  ergs  when  we 
know  the  height  I  in  centimetres 
and  the  force  in  dynes,  which  is  980  times  the  mass  in 
grams.  If  the  body  should  now  fall  towards  the  earth 
a  distance  of  I  centimetres  it  would  possess  an  amount 
of  kinetic  energy  equal  to  the  potential  energy  which 
it  has  lost  in  falling  the  distance  I,  The  body  has  a 


FIG.  116. 


UNITS   OF  ENERGY.     EQUIVALENTS.  163 

definite  velocity  at  any  instant,  as  every  moving  body 
has.  Let  us  try  to  find  an  expression  for  the  energy 
in  terms  of  mass  and  velocity.  The  velocity  of  a  body 
which  has  fallen  for  t  seconds  under  a  constant  accel- 

eration, a,  is 

v  =  at 


If  it  started  from  a  position  of  rest  its  initial  velocity 
was  zero.     Its  average  velocity,  v',  is  therefore, 


But  the  distance,  Z,  traversed  in  time  t  is 
(1)     I  —  v't  =  i  at  -  1  =  i  a£2 

The  force  acting  upon  the  body  was 

(4)     /  —  ma 
and  the  work  done  in  lifting  it  was 

(16)     w  =  /Z 
which  represents  the  energy  it  'has  gained  in  falling. 

w  =  /J  =  mal  = 


Since  at  •=.  v,  a?t2  =  v2,  and 
w  =  fl  =  \  mv2 


That  is  to  say,  the  kinetic  energy  of  a  body  of  mass  w, 
moving  with  a  velocity  v,  is  measured  by  half  the  prod- 
uct of  its  mass  times  the  square  of  its  velocity. 

A  problem  which  engaged  the   attention  of  several 
physicists    during   the    first   half  of    the    century   just 


164 


WOEK.  ENERGY.  MACHINES. 


closing  was  the  determination  of  the  mechanical  equiva- 
lent of  heat,  or,  as  we  may  now  express  it,  the  measure- 
ment of  the  quantity  of  energy  represented  by  a  defi- 
nite quantity  of  heat.  Dr.  Joule,  of  England,  made  the 
determination  by  means  of  a  calorimeter  in  which  a 
known  mass  of  water  was  heated  by  the  motion  of  a 

paddle  wheel  actuated 
by  falling  weights  (see 
Fig.  117).  The  rise  in 
temperature  times  the 
mass  of  the  water  meas- 
ured the  heat.  He  found 
the  work  represented 
by  one  calorie  of  heat 
to  be  41,590,000  ergs. 
The  factor  4.159  XlO7 
is  known  as  Joule's 
equivalent  and  is  represented  by  J.  The  value  of  J 
from  the  most  recent  determinations  is  4.19  X  107.  We 
may  express  calories  in  ergs  by  multiplying  by  J. 

(21)     w=JH 

The  energy  liberated  or  consumed  during  any  chemi- 
cal change  may  be  measured  by  allowing  the  change  to 
take  place  in  a  calorimeter  and  measuring  the  quantity 
of  heat  evolved  or  absorbed.  In  like  manner,  the 
energy  of  the  electric  current  may  be  measured  and 
compared  with  the  energy  supplied  in  the  form  of  me- 
chanical motion  to  the  dynamo  which  generates  the 
current. 


FIG-  m- 


UNITS   OF  ENEEGY.     EQUIVALENTS.  165 

Radiant  energy,  from  the  sun  or  any  hot  body,  may 
be  measured  by  allowing  the  radiations  to  fall  upon  a 
surface  wliicli  absorbs  the  radiations  and  converts  them 
into  heat.  Radiant  energy  consists  of  a  wave  motion 
which  is  transmitted  through  the  ether.  It  becomes 
known  to  us  only  after  it  is  converted  into  some  other 
form  of  energy,  as  when  in  the  form  of  light  it  affects 
our  eyes  (probably  a  chemical  effect),  or  the  sensitive 
photographic  plate  (certainly  a  chemical  effect),  or 
when  it  takes  the  form  of  heat,  as  all  forms  of  energy 
are  prone  to  do. 

We  shall  discuss  the  nature  of  radiant  energy  more 
fully  in  a  later  chapter.  For  our  present  purposes  it  is 
enough  to  know  that  it  is  a  form  of  energy  which  is 
transmitted  at  enormous  velocities  through  what  we 
call  empty  space  as  well  as  through  matter,  and  that 
it  may  be  transformed  directly  into  heat  or  chemical 
energy  and  thus  indirectly  into  other  forms. 

To  recapitulate,  we  are  accustomed  to  distinguish 
five  forms  of  energy:  (a)  Mechanical  energy,  which  is 
kinetic  in  a  moving  body,  potential  in  a  body  which 
has  been  moved  out  of  a  position  of  stable  to  a  posi- 
tion of  unstable  equilibrium ;  (6)  thermal  energy, 
which  is  kinetic  when  it  raises  the  temperature  of 
bodies,  potential  when  it  changes  the  state  from  solid 
to  liquid  or  liquid  to  gaseous ;  (c)  chemical  energy, 
which  is  kinetic  in  the  form  of  heat  or  the  electric 
current  at  the  instant  the  chemical  change  takes  place, 
but  is  potential  in  all  compounds  which  are  not,  like 


166         WORK.  ENERGY.  MACHINES. 

the  oxides,  in  the  most  stable  state  possible ;  (c?)  elec- 
trical energy,  which  is  potential  in  the  form  of  the  static 
charge  and  in  the  magnet,  but  kinetic  in  the  current ; 
(e)  radiant  energy,  which  is  always  kinetic  and  is 
therefore  the  form  which  all  energy  tends  to  take,  for 
this  is  the  law  of  all  change.  Bodies  which  are  in  an 
unstable  condition  tend  to  lose  the  potential  energy 
which  they  possess  by  virtue  of  their  instability.  For 
example,  all  masses  which  are  lifted  tend  to  fall;  all 
springs  which  are  bent  tend  to  straighten ;  all  gases  tend 
to  liquefy  and  all  liquids  tend  to  condense ;  all  ele- 
ments tend  to  oxidize  or  to  undergo  a  change  similar 
to  oxidation ;  all  electric  charges  tend  to  discharge. 
All  of  the  changes  mentioned  generate  heat  which  radi- 
ates off  into  space,  and  if  the  loss  were  not  supplied 
by  the  radiant  energy  which  comes  to  us  from  the  sun 
our  earth  would  soon  become  cold,  motionless,  and  life- 
less. 

1 06.  Energy  and  Life.  —  Why  do  living  beings 
work,  why  must  we  work  to  live  at  all,  and  work  much 
to  live  as  we  want  to  live  ?  Our  food,  at  once  the  ma- 
terial of  which  our  bodies  are  built  and  the  source  of 
the  energy  of  all  our  bodily  movements,  does  not  lie 
within  our  grasp.  It  must  be  sought,  often  pursued ; 
when  got  it  must  be  prepared.  Our  clothing,  houses, 
implements,  furniture,  must  be  gathered  in  crude  form 
and  shaped  to  meet  our  uses,  must  be  converted  from 
its  present  condition  to  a  less  stable  one,  must  have 


SOURCES   OF   USEFUL  ENERGY.  167 

work  clone  upon  it  to  fit  it  for  our  use.  The  earth 
which  covers  the  stone  in  the  quarry  must  be  thrown 
aside,  the  stone  must  be  broken  in  pieces,  lifted  out 
and  carried  to  a  new  location,  shaped  Avith  the  chisel, 
set  in  place,  before  it  is  a  part  of  the  foundation  of  a 
house.  Every  one  of  these  operations  was  resisted  by 
a  force,  gravitation  or  cohesion,  which  opposed  the 
change  and  necessitated  the  doing  of  work. 

The  man  who  shapes  a  tool  of  wood  or  iron  must 
pull  apart  with  knife  or  file  the  molecules,  or  bend  and 
shape  them  into  new  forms.  All  these  changes  are 
opposed  by  forces  to  overcome  which  requires  the  doing 
of  work. 

As  man's  wants  and  desires  increase  he  finds  that 
more  and  more  work  is  required  if  he  is  to  satisfy  them. 
The  amount  of  energy  which  his  own  body  can  supply 
is  limited.  The  civilized  man  can  assimilate  no  more 
food  than  the  savage.  If  he  is  to  do  the  things  he 
desires  to  do,  he  must  find  other  sources  of  energy 
which  he  may  utilize.  They  lie  ready  to  his  hand  if 
only  his  brain  can  devise  ways  to  use  them. 

107.  Sources  of  Useful  Energy.  —  Among  the 
sources  of  energy  available  for  man's  use  are  (a)  the 
bodily  energy  of  animals;  (5)  the  energy  of  flowing 
water  and  the  tides  and  waves  of  the  sea;  (<?)  the 
energy  of  the  winds ;  (c?)  chemical  energy,  particularly 
of  combustion  and  of  electric  batteries. 

Let  us  trace  these  secondary  sources  of  energy  back 


168  WORK.     ENEEGY.     MACHINES. 

as  far  as  possible  to  see  if  any  of  them  have  a  common 
source.  The  bodily  energy  of  animals  is  derived  from 
the  food  they  eat,  which  consists  of  other  animals  and 
of  plants.  The  animals  eaten  lived  on  plants,  so  that 
ultimately  the  source  of  food  supply  is  almost  wholly 
the  plant  kingdom.  Now  plants  are  able  to  use  the 
stable  compounds  of  the  soil  and  air  and  convert  them 
into  unstable  compounds  useful  for  animal  food  only 
under  the  influence  of  sunlight.  The  sun  is  then  the 
ultimate  source  of  animal  energy.  The  water  which 
drives  our  mills  does  so  because  it  is  in  a  position  to 
flow  down  hill.  It  possesses  potential  energy  because 
it  has  been  lifted  to  the  clouds  in  the  form  of  vapor  by 
heat  derived  from  solar  radiation,  and  it  delivers  its 
energy  to  any  one  who  can  use  it  on  its  way  back  to 
the  sea.  The  winds,  too,  are,  as  we  saw  when  studying 
heat,  convection  currents  in  the  air  set  up  by  heat 
received  from  solar  radiation.  The  waves  of  the  sea 
are  due  chiefly  to  the  winds,  hence  windmills  and  wave 
mills  are  driven  indirectly  by  solar  radiation. 

The  energy  of  expanding  steam  which  we  utilize  in 
the  steam  engine  we  derive  from  the  chemical  combina- 
tion of  the  carbon  of  wood  and  coal  with  the  oxygen 
of  the  air.  Combustion  is  in  fact  a  rapid  oxidation  of 
carbon  or  hydrogen.  The  coal  was  once  living  trees 
into  whose  composition  the  carbon  was  wrought  after 
-being  torn  from  its  stable  union  with  oxygen  by  the 
energy  supplied  to  the  leaves  of  the  plants  by  solar 
radiation. 


MACHINES.  169 

Chemical  batteries  employ  substances,  like  zinc  and 
acid,  which  are  not  found  in  their  present  condition  in 
nature  but  have  had  work  done  upon  them  by  man. 
Batteries  ought  not  to  be  reckoned  therefore  as  primary 
sources  of  energy.  The  tides  are  due  to  the  interac- 
tion of  the  earth  and  the  moon,  and  do  not  therefore 
derive  their  energy  from  a  distance.  We  see,  however, 
that  the  sources  of  energy  upon  which  we  depend 
chiefly,  and  indeed  almost  wholly,  are  all  traceable  to 
one  source  —  the  sun.  When  we  have  used  this  energy 
or  let  it  pass  us  by  unused,  as  by  far  the  greater  part 
does  and  always  will,  it  passes  out  again  into  that 
boundless  ocean  of  ether  whence  it  came. 

1 08.  Machines.  —  Man  alone,  of  all  animals,  uses 
tools,  implements,  machines.  In  a  broad  sense  a 
machine  may  be  defined  as  a  device  by  means  of  which 
energy  is  transformed  so  as  to  be  advantageously 
applied.  A  man  could  dig  a  well  (in  some  localities) 
with  his  hands  alone,  but  give  him  a  shovel  with  which 
he  can  pry  the  dirt  apart  and  throw  it  out  twenty  hand- 
fuls  at  a  time  and  he  will  work  to  better  advantage 
than  without  the  simple  machine.  If  the  well  is  deep 
he  will  gain  an  additional  advantage  if  he  use  a  rope 
and  bucket  to  lift  the  clods  to  the  top.  He  may  even 
find  it  profitable  to  attach  a  windlass  to  his  rope  and 
use  a  much  larger  bucket  than  he  could  lift  without  it. 
A  horse  can  carry  a  man  or  a  small  load  on  his  back ; 
hitched  to  a  wagon  on  good  roads  he  can  draw  ten 


170 


WORK.     ENERGY.     MACHINES. 


times  the  load.  Sewing  by  hand  goes  slowly  because 
the  limit  of  speed  is  reached  far  sooner  than  the  limit 
of  strength.  The  sewing  machine  transforms  strength 
into  speed  and  multiplies  the  power  of  the  seamstress 
sevenfold. 

In  our  study  of  machines  we  shall  perhaps  proceed 
in  the  order  of  simplicity  if  we  follow  the  historical 
order  and  take  up  first  the  traditional  simple  machines. 

109.  The  Lever. — A  rigid  bar  arranged  to  turn 
about  a  fixed  point  is  called  a  lever.  The  fixed  point 
is  called  the  fulcrum  (see  Fig.  118).  The  two  forces  are 

applied  at   different 

P'  O  P      points  on  the  bar,  P, 

Pr.     By  the  law  of 
moments   there  will 
be  equilibrium  when 
the    force  f    which 
acts  at  P  times  the 
distance  P  0  is  equal 
to  f  times  P'  Q,  and 
the  moments  are  op- 
posite   in    direction. 
Both  forces  may  act 
on  the  same  side  of 
the  fulcrum,  only  so  that  their  moments  are  opposite. 
Let  us  now  suppose  the  moment  of  the  force  /  to  be  a 
little  greater  than/7,  so  that 
/  X  PO  —  /'  X  P'O  +  the  friction  of  the  bearing 


FIG.  118. 


THE  LEVER.  171 

There  will  then  be  motion  in  the  direction  PQ.  Then 
the  work  done  by  the  force  f  in  moving  the  distance 
PQ  =  I  is  fl  and  the  work  done  at  P1  is  f'l'.  Since  I 
and  V  are  arcs  of  circles  whose  radii  are  PO  and  P1  0 
respectively,  it  follows  from  geometry  that 

PO         I 
(fl)    1*0  =  T 

But  from  the  law  of  moments 

(i)    fXPO=fxP'0     or     jr  = 
Whence  it  follows  from  (#)  and  (6)  that 


fl=flf 

when  the  forces  are  in  equilibrium.     When  motion  is 
produced,  /£  =  f'l'  -\-  work  done  in  overcoming  friction, 

w",  whence 

(17)     w  =  w'  +  w" 

which  is  the  law  of  work. 

In  the  older  treatises  on  mechanics  the  force  applied, 
/,  was  called  the  power  and  the  force  obtained,  f,  was 
called  the  iveight.  The  words  power  and  weight  have 
very  definite  meanings  in  physics  and  meanings  not 
at  all  alike.  It  seems  best  not  to  retain  these  words, 
therefore,  in  our  discussion  of  machines. 

The  distances  PO  and  P1  0  being  straight  lines  are 
easier  to  measure  than  the  arcs  PQ  and  P'Q.  Their 
ratio  may  be  used  therefore  for  the  ratio  of  I  to  I',  to 
which  it  is  always  equal. 


172  WORK.    ENERGY.     MACHINES. 

The  lever  has  numerous  applications.  The  spade, 
the  handspike  or  crowbar,  the  pump  handle,  all  serve 
to  enable  a  man  to  overcome  forces  greater  than  he  is 
able  to  exert,  but  always  at  the  expense  of  distance. 
In  every  case  the  hand  moves  a  greater  distance  than 
the  object  at  the  other  end  of  the  lever. 
f 

The  ratio  —^-  is  called  the  mechanical  advantage  of  the 

machine.  It  is  to  be  borne  constantly  in  mind  that  the 
real  advantage  consists  in  obtaining  the  energy  in  a 
form  better  suited  to  our  needs.  The  traveller  who  gets 
four  marks  for  a  dollar  in  Germany  or  five  francs  for  a 
dollar  in  France  is  pleased "  because  he  gets  what  he 
needs,  and  he  is  even  willing  to  pay  the  broker  a  small 
percentage  for  making  the  exchange.  In  mechanical 
transfers  we  must  always  pay  exchange  too.  It  is  in 
the  form  of  heat  and  is  always  delivered  on  the  spot. 
We  make  it  as  small  as  possible  by  avoiding  friction 
and  other  sources  of  heat  waste,  but  no  rational  me- 
chanic expects  to  get  back  all  he  puts  in,  much  less  win 
any  energy  by  means  of  a  machine. 

A  machine  which  transfers  energy  with  but  little  loss 
is  said  to  be  an  efficient  machine,  and  the  ratio  of  w' 
to  w  is  called  the  efficiency  of  the  machine.  Since 
w  =.  wr  -\-  w"  and  since  wff  is  never  zero,  it  follows  that 
w' /w  is  always  less  than  unity,  or,  as  usually  expressed, 
less  than  100  per  cent. 

EYK  •  _  work  obtained 

work  expended 


THE  LEVEE.  173 


W  W 

where  w"  represents  the  total  work  not  obtained  in  use- 
ful form  but  lost  —  usually  in  overcoming  resistance  in 
the  machine  itself.  The  useful  work  obtained,  w',  is 
spent  in  overcoming  resistance  on  objects  other  than  the 
machine,  and  may  take  any  of  the  well-known  forms  of 
kinetic  or  potential  energy. 

The  efficiency  of  a  lever  depends  very  much  on  the 
amount  of  friction  at  the  fulcrum,  but  very  much  more 
upon  the  direction  of  application  of  the  force.  When 
a  lever  rests  upon  a  knife  edge,  as  in  the  balance,  fric- 
tion for  small  loads  may  be  made  exceedingly  small  in 
amount,  but  when  a  similar  arrangement  is  used  with 
the  crowbar  there  must  be  friction  enough  to  keep  the 
bar  from  slipping.  The  use  of  the  crowbar  is  limited 
by  the  fact  that  as  soon  as  the 
object  has  been  lifted  a  short 
distance,  it  must  be  supported 
while  the  fulcrum  is  raised 
so  that  the  operation  may  be 
repeated. 

Where  pressure  is  applied 
from  a  single  direction  while 
the  lever  is  allowed  to  make  FIG- 

a  full  revolution,  as  in  the  crank  of  a  bicycle,  there  is 
only  a  small  part  of  the  revolution,  near  P  (Fig.  119), 
where  the  whole  of  the  force  applied  produces  rotation. 
At  Q  the  component  be  producing  rotation  is  smaller 


174 


WORK.     ENERGY.     MACHINES. 


than  the  component  06,  which  is  wasted  in  producing 
pressure  on.  the  bearing.  At  P  the  efficiency  is  nearly 
100  per  cent,  \vhile  at  Q  it  is  less  than  50  per  cent,  and 
at  the  top  and  bottom  the  force  applied  is  almost  all 
wasted.  We  overcome  this  defect  in  a  large  measure 
by  applying  our  force  mainly  during  that  part  of  the 
revolution  where  it  will  accomplish  most. 

1 10.  The  Pulley.  —  If  the  force  were  applied  to  a 
lever  by  means  of  a  rope  acting  downward  at  the  end 
-  -  of  the  lever  we  should  encounter  the 
same  disadvantages  as  in  the  bicycle 
crank,  but  if  instead  of  a  lever  we  use 
a  grooved  wheel  over  which  a  rope 
passes,  the  force  may  always  be  applied 
at  the  proper  point,  while  the  fulcrum 
may  be  placed  once  for  all  higher  than 
the  point  to  which  the  load  is  to  be 
lifted.  If  the  wheel  be  thought  of  as 
having  spokes  (see  Fig.  120)  it  is  evident 
that  the  horizontal  pair  of  spokes  may  be 
considered  the  two  equal  arms  of  a  lever 


CD 

FIG.  120. 


I  at  the  instant  when  the  wheel  is  in  the 
position  shown.  Since  the  wheel  is  prac- 
tically solid  any  diameter  may  be  such  a  lever.  Friction 
keeps  the  rope  in  position  on  the  wheel  without  greatly 
retarding  the  motion.  The  pulley  is  free  from  the 
defects  of  the  lever,  but  it  offers  in  its  simple  form  no 
mechanical  advantage  whatever.  Since  1=1',  f  must 


THE  PULLEY. 


175 


equal  f  and   more.     The  pulley  even  in   tins   simple 
form  is  useful,  however,  since  it  changes  the  direction 

of  the  application  of  the  force.     A  man     , Q ~ 

can  lift  a  large  bucket  of  water  from  a 
well  more  easily  by  pulling  downward 
on  the  rope  than  by 
leaning  over  the  well 
and  pulling  upward, 
while  a  horse  cannot 
conveniently  pull  in 
any  direction  but  the 
horizontal.  The  two 
fixed  pulleys  used  in 
the  pile  driver  (Fig. 
115)  enable  us  to  em- 
ploy a  horse  for  lift- 
ing the  weight. 


f 


f 


FIG.  121. 

By  the  arrangement 
of  pulleys  shoAvn  in  Fig.  121  the  load 
which  is  attached  to  a  movable. pulley 
moves  but  half  as  far  as  the  point  of 
application  of  the  force  moves,  and 
we  have  a  mechanical  advantage  of  2, 
while  the  use  of  a  third  pulley  as  in 
Fig.  122  gives  us  a  mechanical  advan- 
tage of  3. 
FIG.  122. 

in.  The  Wheel  and  Axle.  —  The  winch  or  wind- 
lass, the  capstan,  the  wheel  and  axle  combine  the  prin- 
ciples of  the  lever  and  the  pulley.  They  all  consist  of 


176 


WORK.    ENEEGY.     MACHINES. 


FIG.  123. 


an  axle  about  which  a  rope  is  wound  drawing  the  load. 

The  load  is  moved  by  force  applied  at  the  circumfer- 
ence of  a  larger 
wheel  or  at  the 
end  of  a  crank. 
Fig.  123  shows 
a  common  winch 
or  windlass.  In 
machine  shops 
combinations  of 
pulleys  of  vari- 
ous sizes  on  the 

same  shaft,  connected  by  belts  with  pulleys  on  various 

other  machines  in  use,  make  it   possible    to    run   any 

machine  in  a  shop 

at    any   desired 

speed  from  an  en- 
gine whose  speed  is 

constant.    When  it 

is  desirable  to  run 

the  same  machine 

at  different  speeds, 

as  is  the  case  with 

the    lathe,    a    pair 

of   cone   pulleys 

is  used.    See  (7,  Or 

(Fig.  124),  where 

D  is  the  driving  pulley  of    the  engine,  SS1  a  line    of 

shafting  from  which  attachments  are  made  either  directly 


FIG.  124. 


THE    WHEEL  AND  AXLE.  177 

to  other  machines  or  to  countershafts  from  which  the 
energy  is  conveyed  to  the  machines  as  wanted.  With 
the  belt  on  the  cone  pulleys  as  shown,  the  lathe  has  its 
lowest  speed.  The  cones  are  of  such  dimensions  that 
the  same  length  of  belt  will  fit  in  all  cases. 

When  two  shafts  are  connected  by  pulley  and  belt 
the  one  carrying  the  larger  pulley  must  go  the  more 
slowly,  since  all  parts  of  the  belt  travel  at  the  same 
speed,  and  a  point  in  the  circumference  of  either  pulley 
travels  with  the  speed  of  the  belt.  In  Fig.  124,  for 
example,  the  driving  wheel,  D,  has  a  speed  of  2,000 
revolutions  per  minute,  and  a  diameter  of  120  inches; 
P  has  a  diameter  of  12,  Q  of  24,  R  of  24,  T  of.  5,  and 
T'  of  20.  To  find  the  speed  of  0'  we  have  only  to 
multiply  2,000,  the  speed  of  the  driving  pulley  D,  by  the 
ratios  of  the  sizes  of  the  successive  pairs  of  pulleys,  thus : 

120       £A          5 
Speed  of  C"  =2,000  X  ~  X  g  X   ^  =5,000 

Where  great  compactness  is  desired  the  two  wheels 
are  brought  into  direct  contact  by  cogs.  A  series  of 
such  cogwheels  is  called  a  train  of  wheels.  By  a 
proper  combination  of  sizes  the  train  of  wheels  in  a 
clock  is  made  to  drive  the  hour,  minute,  and  second 
hands  at  the  proper  relative  speeds.  In  cases  like  the 
bicycle,  where  a  belt  would  slip  and  the  wheels  cannot 
conveniently  be  placed  in  contact,  sprocket  wheels  and 
chain  accomplish  the  purpose.  The  rear  wheel  may  be 
given  any  desired  speed  by  suitably  choosing  the  rela- 
tive size  of  the  two  sprocket  wheels. 


178 


WORK.     ENERGY.     MACHINES. 


ii2.  The  Inclined  Plane.  —  When  a  body  is  rest- 
ing  on  an  inclined  plane  its  weight  is  supported  partly 
by  the  plane,  partly  by  some  force  acting  parallel  to  the 
plane.  The  ratio  of  the  two  forces  is  the  ratio  of  the 
two  components,  be,  ac  (Fig.  125). 
The  ratio  of  the  force  overcome 
to  the  force  applied  is 

ba 
A 


ac 


FIG.  125.  .  J 

But  since  the  angles  B  and  b  have  their  sides  mutually 
perpendicular,  triangles  ABO  and  abc  are  similar  and 
ba         BA  [ 


Whence  it  follows  that 

<->  £ = 

that  is  to  say,  the  advantage  of 
an  inclined  plane  is  the  ratio 
of  the  length  to  the  height  of 
the  plane,  provided  that  the 
force  acts  along  the  plane. 

The  wedge  may  be  treated  as 
a  pair  of  inclined  planes  placed 
base  to  base  (see  Fig.  126).  The  force  is  applied  in  the 
direction  OB  to  overcome  the  force  of  cohesion  which 
is  in  a  direction  at  right  angles  to  OB.  It  follows  that 
'  f>  _  GB 

'~ 


FIG.  126. 


THE  SCREW. 


179 


When  wedges  are  used  to  split  large  stones  a  row  of 
holes  is  drilled  along  the  line  on  which  the  stone  is  to 
be  divided  and  several  wedges  are  driven  at  the  same 
time.  If  the  only  object  aimed  at  is  to  shatter  the 
stone  a  charge  of  powder  or  dynamite  is  often  used. 

113.  The  Screw.  —  When  a  wagon  road  or  a  rail- 
way is  to  he  carried  up  a  very  steep  mountain  it  does 
not  go  straight 
up,  but  winds 
in  a  spiral, 
making  the 
ascent  by  go- 
ing several 
times  the  dis- 
tance. The 
length  of  the 
incline  is  thus 
increased  with- 
out increasing 
its  height,  thus 
greatly  increasing  the  mechanical  advantage.  The  same 
principle  is  employed  in  the  screw,  where  the  advantage 
is  usually  still  further  increased  by  the  use  of  a  wheel 
or  a  lever  at  the  head  of  the  screw  (see  Fig.  127). 
The  distance  which  the  screw  advances  in  one  revolution 
is  the  distance,  d,  between  the  adjoining  threads  of  the 
screw  measured  parallel  to  the  axis  of  the  screw,  while 
the  distance  moved  by  the  applied  force  is  the  circum- 


FlG.  12 


180        WORK.  ENERGY.  MACHINES. 

ference,  <?,  described  by  the  point  of  application  of  the 
force.     The  advantage  of  the  screw  is  therefore 


Thus  a  force  of  2  pounds  applied  to  the  handle  of  a 
screw-driver  2.1  inches  in  diameter  (=  6.6  inches  in  cir- 
cumference) w^ould  exert  upon  a  screw  having  12 
threads  to  the  inch  a  force  of  158.4  pounds. 

The  simple  machines  just  described  —  the  lever,  the 
wheel,  the  incline  (including  the  screw)  —  have  been 
known  and  used  for  thousands  of  years.  The  innu- 
merable mechanical  contrivances  in  use  to-day,  from  the 
wheelbarrow  to  the  typesetting  machine,  are  all  but 
combinations  of  the  simple  machines.  It  is  when  we 
seek  to  employ  other  forms  of  energy  than  those 
known  to  the  ancients  (animals,  wind,  waterfalls)  that 
we  meet  machines  involving  principles  not  found  in 
the  simple  machines,  but  always  associated  in  practice 
with  them. 

114.  Heat  Engines.  —  As  long  ago  as  B.C.  200, 
machines  employing  the  expansive  force  of  heated  air 
and  steam  were  described,  by  Hero,  or  Heron,  of  Alex- 
andria. He  arranged  the  device  shown  in  Fig.  128, 
by  which  a  fire  kindled  on  the  hollow  altar  causes  the 
air  within  to  expand,  forcing  the  water  out  through  the 
siphon  into  a  bucket,  where  its  weight  served  to  open 
the  temple  doors.  When  the  fire  was  allowed  to  die  out 
the  doors  closed  as  miraculously  as  they  had  opened. 


HEAT  ENGINES. 


181 


Hero's  steam  engine  is  shown   in   Fig.  129.      It   is 
easily  constructed.      Steam    from    the    basin  enters  by 
one  of   the  hollow  columns  supporting  the  ball,  which 
is  made  to  revolve  by 
the    reaction   of    the 
air  upon  the  escaping 
steam. 

The  steam  engine 
invented  by  Watt  in 
1765  was  the  out- 
grow t  h  of  various 
forms  of  the  "  atmos- 
pheric "  engines  which 
were  the  fruit  of 
Guericke's  invention 
of  the  air  pump  a 
hundred  years  before. 
In  these  engines  a  vacuum  was  produced  in  a  cylinder 

by  the  condensation  of  steam 
within  it,  and  a  piston  was 
?/;  V'  pushed  into  the  cylinder  by 
the  pressure  of  the  atmos- 
phere. Such  engines  were 
used  for  pumping  water  from 
mines,  but  were  very  waste- 
ful of  steam. 

Watt  not  only  invented  the 
steam     engine,    which    with 
Fl(;  129_  slight   modifications    is  used 


FIG.  128. 


182 


WOEK.     ENERGY.     MACHINES. 


to-day,  but  he  also  invented  most  of  the  devices  by 
which  it  is  controlled  and  rendered  efficient,  such  as 
the  automatic  governor  and  .throttle  valve.  The  slide 
valve  was  invented  by  his  assistant,  Murdoch. 

The  principle  of  the  steam  engine  may  be  under- 
stood from  Fig.  130.  The  steam  from  the  boiler  enters 
the  steam  chest  at  $  whence  it  is  admitted  to  the 
cylinder  through  the  ports,  P,  P',  by  means  of  the 


FIG.  iso. 

slide  valve,  V,  which  opens  port  P  while  the  piston  is 
near  the  end  J.,  at  the  same  time  closing  the  passage 
from  P'  to  the  steam  chest  and  opening  it  to  the  ex- 
haust, thus  allowing  the  condensed  steam  to  escape 
at  atmospheric  pressure.  By  the  time  the  shaft  has 
made  half  a  revolution  the  piston  is  in  suitable  position 
at  the  other  end  of  the  cylinder,  the  valve  has  moved 
so  as  to  open  P',  arid  live  steam  is  now  admitted  to  the 
right-hand  side  of  the  piston  head,  while  condensed 
steam  escapes  through  P  to  the  air. 


HEAT  ENGINES. 


183 


The  oscillating  motion  of  the  piston  is  converted 
into  rotary  motion  by  a  crank  attached  to  a  shaft  which 
carries  a  large  driving  pulley.  Attached  to  the  same 
shaft  is  a  smaller  crank  or  eccentric  which  operates  the 
slide  valve. 

In  the  best  steam  engines  not  more  than  twenty  per 
cent  of  the  energy  liberated  in  the  combustion  of  coal  is 
utilized,  yet  coal  is  so  cheap  that  it  continues  to  furnish 
most  of  the  energy  used  for  manufacturing  purposes. 

115.  A  form  of  engine  which  is  coming  quite  gen- 
erally into  use  where  but  a  small  amount  of  power  is 
required  is  the  so-called  gas  engine  (Fig.  131).  The 
propulsive  force  is 
furnished  by  the  ex- 
plosion produced  by 
igniting  a  mixture  of 
air  and  coal  gas  or  gas- 
oline vapor  within  the 
cylinder.  The  gas 
mixture  is  admitted 
only  at  every  other  rev- 
olution, since  the  prod- 
ucts of  combustion  must  first  be  expelled  by  the 
piston  on  its  first  return  stroke.  During  the  second 
stroke  the  mixed  gases  are  admitted  through  a  valve, 
which  closes  like  a  pump  valve  when  the  piston  starts 
back.  When  the  piston  is  at  the  end  of  its  stroke  and 
has  compressed  the  gases  it  closes  an  electric  circuit, 


PIG.  i3i. 


184        WORK.  ENERGY.  MACHINES. 

which  is  broken  when  the  piston  starts  on  its  second 
outward  stroke,  producing  a  spark  which  ignites  the 
gases,  and  the  cycle  of  operations  is  then  repeated. 

The  fact  that  the  force  is  exerted  on  but  one  side  of 
the  piston  and  but  once  in  two  revolutions  makes  the 
engine  less  steady  than  the  steam  engine,  with  its  two 
impulses  for  each  revolution.  This  fault  is  overcome 
to  some  extent  by  the  use  of  heavy  fly  wheels. 

Regulation  is  accomplished  by  a  governor,  which 
entirely  closes  the  supply  valve  when  the  speed  exceeds 
a  certain  amount.  When  the  load  is  light  the  explo- 
sions occur  only  at  rare  intervals,  so  that  but  little  gas 
is  wasted.  Gas  engines  are  now  used  to  drive  self- 
propelling  carriages,  a  fact  which  is  likely  to  lead  to  a 
very  light  and  compact  form  of  machine.  At  the 
present  time  it  is  more  expensive  than  the  horse  as 
regards  first  cost,  if  not  as  regards  maintenance. 

The  gas  engine  has  a  higher  efficiency  than  the  steam 
engine,  yet  it  has  not  nearly  reached  the  highest  effi- 
ciency of  which  it  is  theoretically  capable. 

116.  Rate  of  Doing  Work.  Power.  Activity.  — 
The  amount  of  energy  which  any  machine  transforms 
in  unit  time  is  its  rate  of  doing  work,  or,  briefly,  its 
power  or  activity.  Watt  introduced  the  term  horse- 
power as  the  unit  of  activity,  and  defined  it  as  33,000 
foot  pounds  per  minute.  The  metric  unit  is  the  watt. 
It  is  defined  as  10,000,000  ergs  per  second  =  1  joule 
per  second  =  1  volt  X  1  ampere. 


OTHEE  ENGINES.  185 

(29)     1  h.p.=  746  watts 

The  horse-power  of  a  dynamo  is  usually  expressed 
in  kilowatts.  Thus  a  50  k.w.  dynamo  would  furnish 
50  amperes  at  1,000  volts  and  would  require  a  67  h.p. 
engine  to  run  it.  If  we  now  recall  that  a  calorie  is 
4.19  joules,  we  see  how  easy  it  is  to  measure  both 
energy  and  power  in  any  one  of  its  forms,  and  then  to 
compare  the  quantity  measured  with  the  same  quantity 
measured  in  another  form.  This  enables  us  to  find  the 
efficiency  of  a  machine  like  the  heat  engine,  which 
transforms  energy  from  one  form  to  another. 

117.  Other  Engines.  —  The  applications  of  the  laws 
of  energy  and  the  principles  of  machines  to  water 
wheels,  tide  wheels,  and  windmills  may  be  left  as  ari 
exercise  for  the  ingenuity  of  the  student  as  he  may  have 
opportunity  to  study  them.  He  has  only  to  remember 
that  no  exception  has  ever  been  found  to  the  law  of 
the  conservation  of  energy,  and  he  may  feel  very  sure 
that  the  man  who  claims  to  have  invented  a  machine 
which  will  run  and  do  work,  even  enough  to  overcome 
friction,  after  the  energy  supplied  to  it  has  been  ex- 
pended, is  either  deceived  himself  or  is  trying  to 
deceive  others  for  his  own  profit.  The  man  who  is 
pursuing  the  phantoms  of  perpetual  motion  should 
devote  himself  to  solving  some  of  the  many  problems 
which  offer  sure  reward  to  the  successful  inventor. 

No  man  can  hope  to  create  energy.  No  man  need 
wish  to  do  so,  for  energy  by  the  millions  of  horse-power 


186        WOEK.  ENERGY.  MACHINES. 

is  going  to  waste  constantly  at  our  very  doors.  The 
energy  contained  in  the  water  which  falls  on  the  roof 
of  a  large  hotel  in  a  year  would,  if  utilized,  run  the 
elevator  twelve  hours  a  day  the  year  round.  The 
energy  of  the  sun  which  falls  on  a  large  factory  and  is 
radiated  again  into  space  would,  if  all  utilized,  run  all 
the  machines  in  the  factory.  A  small  fraction  of  the 
energy  of  -the  wind  which  blows  across  every  farm 
would,  if  harnessed,  do  all  the  work  of  the  farm. 

The  men  who  are  to  profit  themselves  and  their  gen- 
eration by  their  inventions  must  first  master  the  known 
laws  of  nature  and  then  apply  them  to  existing 
problems. 

118.  The  Storage  of  Energy. — If  the  energy  from 
so  intermittent  a  source  as  the  wind,  for  example,  is  to 
be  utilized  to  any  large  extent  by  people  who  must 
work  every  day,  means  must  be  found  to  store  the 
excess  of  one  day  against  the  needs  of  another  day. 
On  the  other  hand,  some  forms  of  energy,  like  light,  are 
consumed  principally  at  certain  hours  of  the  day,  and 
the  engine  must  have  power  enough  to  supply  the 
greatest  demand,  though  an  engine  a  third  as  large 
could  do  the  work  if  it  were  distributed  uniformly 
throughout  the  day.  Many  large  lighting  stations  now 
have  large  storage  batteries  which  are  charged  during 
the  middle  of  the  day  and  help  carry  the  load  in  the 
evening.  Indeed,  the  storage  battery  is  as  yet  the  only 
successful  device  for  storing  energy  on  a  large  scale. 


THE    TRANSFERENCE   OF  ENERGY.  187 

Plants,  as  we  know,  store  the  energy  of  the  sun's 
rays,  but  the  supply  thus  stored  could  not  meet  our 
demands  if  the  treasures  stored  in  the  coal  fields  dur- 
ino-  the  ages  rich  in  carbon  were  exhausted.  Ere  the 

o  o 

coal  fields  are  all  used  ways  will  perhaps  be  found  to 
collect  and  store  the  solar  energy  by  artificial  means 
which  will  make  us  independent  of  coal,  as  iron  and 
paper  are  making  us  independent  of  wood,  and  as 
aluminum  might  easily  make  us  independent  of  copper 
and  tin. 

119.  The  Transference  of  Energy  from  Place  to 
Place.  —  It  often  happens  that  large  waterfalls  are 
located  at  some  distance  from  a  mine,  where  it 
would  be  very  desirable  to  use  the  energy  of  the  fall- 
ing water.  The  desired  end  may  be  accomplished  by 
converting  the  energy  into  the  form  of  an  electric  cur- 
rent, when  it  may  easily  be  carried  by  wires  to  a  con- 
siderable distance,  to  be  again  converted  by  means  of 
motors  into  mechanical  motion. 

In  large  towns  there  are  usually  a  number  of  small 
shops,  printing  oifices,  etc.,  which  need  a  small  amount 
of  power  at  somewhat  irregular  intervals.  The  total 
amount  of  power  needed  can  be  supplied  from  one 
large  engine,  with  one  or  two  men  to  care  for  it,  much 
more  economically  than  by  a  number  of  small  engines. 

Several  plans  have  been  used  for  distributing  the 
power.  The  pressure  applied  to  water  by  a  large 
pump  at  a  central  station  may  be  used  by  water  motors 


188  WORK.    ENEBGY.     MACHINES. 

at  any  place  on  the  line  of  pipes.  The  friction  of 
moving  water  on  the  pipes  is,  however,  so  great  that 
very  large  pipes  are  required  both  for  the  inflow  and 
the  escape  of  the  water,  so  that  the  ordinary  service 
pipes  are  not  sufficient  to  furnish  any  considerable 
power  at  the  pressures  usually  employed.  Compressed 
air  furnishes  a  simple  means  of  conveying  power,  but 
one  which  has  not  come  into  general  use  because  of  the 
expense  of  laying  pipes  of  sufficient  strength  and  size. 

Here  again  the  electric  current  is  found  to  be  admi- 
rably adapted  to  the  distribution  of  power.  Electric 
conductors  may  be  carried  anywhere  and  tapped  at  any 
point ;  indeed,  the  point  of  contact  may  be  continuously 
changing,  as  in  the  trolley  car,  without  any  serious  loss 
}yj  leakage.  The  ease  with  which  the  current  may  be 
turned  on  or  off  or  varied  in  amount  at  pleasure,  and 
the  variety  of  uses  to  which  it  may  be  put  —  heat,  light, 
mechanical  work,  electroplating  —  make  the  electric  cur- 
rent the  ideal  form  of  energy  for  transference  to  a  dis- 
tance and  for  distribution  in  small  quantities. 

When  the  energy  is  to  be  carried  long  distances  high 
potentials  are  used,  because  the  heat  lost  in  the  wire  is 
proportional  to  the  square  of  the  current  strength. 
Thus  if  100  kilowatts  are  to  be  carried  at  100  volts,  we 
have 

100  k.w.  =  100  volts  X  100  amperes 

but  at  1,000  volts  we  have 

100  k.w.  =  1,000  volts  X  10  amperes 


EXERCISES.  189 

In  the  first  case  the  current  strength  is  ten  times  that 
in  the  second,  and  the  heat  lost  in  the  wire  is  102  or 
100  times  as  much  for  a  wire  of  a  given  size. 

Exercises. 

89.  (a)  Two  men  lift  a  200  pound  barrel  of  salt  upon  a 
wagon  3  feet  high.  How  much  work  do  they  do  ?  (6)  One 
man  rolls  the  same  barrel  of  salt  upon  the  same  wagon,  using 
an  inclined  plane  9  feet  long.  How  much  force  does  he  exert  ? 
How  much  work  does  he  do  ?  (See  Fig.  132.). 


FIG.  132. 


90.  A  man  weighing  150  pounds  carries  50  pounds  of  brick 
up  a  ladder  20  feet  high,     (a)  How  much  work  has  he  done  ? 
(6)  How   much  useful  work?     (c)  What  is  the    efficiency  of 
this  method  of  working  ?     If  he  can  draw  up  100  pounds  to 
the  same  height  and  in  the  same  time  by  means  of  a  single 
pulley  and  a  box  weighing  20  pounds,  friction  being  equivalent 
to  30  pounds,  (d)  how  much  work  does  he  do  ?    (e)  How  much 
useful  work  ?  (/)  What  is  the  efficiency  of  the  method  used  ? 

91.  A  horse  which  can  exert  continuously  a  force  of   500 
pounds  is  used  to  lift  the  material  for  a  building.     A  number 
of  iron  girders,  each  weighing  600  pounds,  are  to  be  lifted. 
What  combination  of  pulleys  will  just  use  the  full  strength  of 
the  horse  if  20  per  cent  of  the  work  he  does  is  spent  in  over- 
coming friction  ? 

92.  A  certain  class  of  freight  engines  can  pull  a  load  of  100 
tons  up  a  grade  of  2  feet  in  100.     Suppose  all  grades  on  the 
line  reduced  to  1  foot  in  100,  what  load  can  these  engines  pull 


190 


WORK.     ENERGY.     MACHINES. 


if  we  suppose  the  work  done  in  overcoming  friction  equal  to 
i  the  work  done  by  the  engines  ? 

$3.  A  crane  (Fig.  133)  employs  a  system  of  5  fixed  and  5 
movable  pulleys  and  a  windlass  having  an  axle  8  inches  in 
diameter,  to  which  is  attached  a  wheel  having  40  cogs  driven 
by  a  wheel  having  8  cogs,  to  which  is  attached  a  crank  16 
inches  long.  How  large  a  stone  can  a  man  lift  if  he  applies  a 
force  of  25  pounds  to  the  end  of  the 
crank  ?  K"o  allowance  is  to  be  made  for 
friction. 

94.  (a)  How  much  work  is  required 
to  lift  a  ton  (2,000  pounds)  of  coal  from 
a  mine  500  feet  deep  ?    (&)  What  must, 
be  the  power  of  an  engine  to  lift  20 
tons  per  day  from  this  mine  ? 

95.  How  much  coal  would  the  engine 
in  the  last  exercise  consume  per  day  if 
it  converts  15  per  cent  of  the  energy 
of  combustion  of  coal  into  mechanical 
work  ?     The  heat  of  combustion  of  coal 
is  7,800  calories  per  gram. 

96.  A  man  lifts  his  own  weight,  say 
150  pounds,   2   inches    every   step   he 
takes.     What  is  his  rate  of  working  if 

FIG.  133.  ne  takes  steps  2£  feet  in   length   and 

walks  on  a  level  at  the  rate  of  3  miles  per  hour  ? 

97.  A  man  weighing  145  pounds  propels  himself  at  the  rate 
of  10  miles  an  hour  on  a  bicycle  weighing  25  pounds  over  level 
pavement.     If  he  should  remove  his  feet  from  the  pedals  at 
any  instant,  the  wheel  would  run  528  feet  before  being  brought 
to  rest  by  friction.     What  is  the  man's  rate  of  working  ? 

98.  A  storage  battery  is  charged  at  the  rate  of  20  amperes 


EXEBCI8ES.  191 

• 

for   12  hours.     It  will  give  a  current  of   10  amperes   for   20 
hours.     What  is  its  efficiency? 

99.  The  battery  mentioned  in  the  preceding  exercise  has  a 
difference  of  potential  of  50  volts,     (a)  When  discharging  at 
16  amperes,  what  is  its  power  in  watts  ?     (&)  If  a  motor  when 
connected  to  this  battery  consumes  8  amperes  at  60  volts  and 
does  work  at  the  rate  of  £  Jiorse-power,  what  is  the  efficiency 
of  the  motor  ? 

100.  A  freight  train  weighing  100  tons  is  moving  at  the  rate 
of  20  miles  an  hour,     (a)  How  much  kinetic  energy  has  it  ? 
(&)  How  much  energy  would  there  be  lost  if  the  train  were 
brought  to  rest  by  applying  the  brakes  ?     (c)  Suppose  that  a 
dynamo   were  connected  to  the  axle  of  one  of  the  cars  and 
made  to  store  energy  in  a  battery,  or  that  a  pump   similarly 
attached  were  used  to  compress  air  in  a  reservoir,  could  a  part 
of  this  waste  energy  be  utilized  ? 

101.  Explain  why  more  coal  must  be  burnt  to  run  a  local 
train  a  given  distance  than  to  run  a  through  train  of  the  same 
size  the  same  distance. 


CHAPTER   VI. 


VIBRATIONS.     WAVES. 

120.  Vibratory  Motion.  —  Most  bodies  which  have 
not  been  recently  disturbed  are  at  rest  in  that  position 
in  which  they  have  the  least  possible  potential  energy : 

a  ball  hung  from  the  end  of 
a  string  hangs  with  the  string 
in  a  vertical  position,  the  ball 
being  as  near  the  earth  as  it 
can  get ;  a  vessel  of  water  has 
its  surface  horizontal ;  a  dew- 
drop  is  spherical.  If  we  draw 
the  ball  to  one  side  (P,  Fig. 
134)  and  let  it  go,  the  poten- 
tial energy  we  have  thus  im- 
parted to  it  will  be  converted 
into  motion,  but  the  ball  will 
not  stop  in  its  former  position 
of  rest,  since  its  momentum 
will  carry  it  past  0  to  P',  a 
point  almost  as  far  from  0  as  P  is.  After  coming  to 
rest  at  P'  it  will  return  through  0  almost  to  P,  repeat- 
ing the  operation  until  the  work  done  by  it  in  pushing 
aside  the  air  has  consumed  the  energy  given  it  when  it 
was  pulled  aside. 

19? 


VIBEATOEY  MOTION.  193 

If  a  vessel  partly  filled  with  water  be  tilted  as  in 
Fig.  135,  and  then  quickly  restored  to  its  original  posi- 
tion, the  water  at  0  will  alternately  rise  and  fall  between 
P  and  P'  and  finally  come  to  rest  at  0. 

Such  motion  is  called  vibratory  motion.  The  dis- 
tance PO  is  called  the  amplitude  of  vibration.  The 
time  occupied  by  the  particle  in  going  from  P  through 


FIG.  135. 

0  to  P1  and  back  again  to  P  is  called  its  period  of 
vibration,  and  the  number  of  vibrations  made  in  one 
second  is  called  the  rate  of  vibration.  All  vibratory 
bodies  which  have  a  definite  and  constant  period  of 
vibration  are  said  to  have  harmonic  motion.  It  is  called 
harmonic  motion  because  all  musical  sounds  are  due  to 
motion  of  that  sort.  Such  motion  is  indeed  very  com- 
mon because  most  bodies  are  in  a  state  of  comparative 
rest  in  stable  equilibrium,  but  are  frequently  disturbed 
by  small  forces  which  are  able  to  set  them  in  vibration, 
though  not  great  enough  to  move  them  from  their 
places.  Thus  the  boughs  of  a  tree  sway  to  and  fro  in 
every  breeze,  yet  stay  in  one  place  for  scores  of  years. 
Whenever  the  force  of  displacement,  as  likewise  the 
force  of  restitution,  increases  directly  with  the  displace- 
ment, the  motion  will  be  harmonic  and  the  period  of 


194  VIBE  A  TIONS.     WA  VES. 

vibration  constant.  We  can  easily  prove  that  the  force 
required  to  stretch  a  spiral  spring  is  proportional  to  the 
elongation  produced  in  the  spring,  for  the  scale  of  a 
spring  balance  is  a  scale  of  equal  parts. 
If  we  hang  a  small  weight  from  a  light 
spiral  spring,  pull  it  downward  and  re- 
lease it,  it  will  vibrate  with  a  slowly  di- 
minishing amplitude,  but  with  a  constant 
period.  The  hairspring  of  a  watch  (Fig. 
136)  vibrates  in  the  same  way.  Vibrating 
pendulums  and  vibrating  springs  furnish  us,  therefore, 
with  the  means  of  measuring  time. 

121.  The  Pendulum.  —  The  pendulum  furnishes  a 
good  example  of  harmonic  motion.  The  force  of  resti- 
tution, that  is,  the  force  tending  to  make  the  ball 
return  to  its  position  of  equilibrium,  is  gravity.  It  is 
obvious  that  when  the  ball  (see  Fig.  137)  is  anywhere 
except  at  0,  gravity  may  be  resolved  into  two  compo- 
nents, one  of  which  acts  to  restore  the  ball  to  its  posi- 
tion of  equilibrium.  This  component  is  greater  at  P 
than  at  Q.  The  force  of  restitution  is  greater  the  greater 
the  amplitude.  Since  there  is  always  a  force  acting 
toward  0,  the  motion  is  accelerated  toward  0,  that  is  to 
say,  the  ball  will  go  faster  and  faster  as  it  approaches 
0,  but  slower  and  slower  as  it  leaves  0.  The  energy 
of  the  pendulum  at  P  and  at  Pf  is  all  potential,  while 
at  0  it  is  all  kinetic.  At  intermediate  points  it  is 
partly  potential  and  partly  kinetic. 


THE  PENDULUM. 


195 


Galileo  noticed  that  the  great  chandelier  in  the 
cathedral  at  Pisa  seemed  to  swing  in  equal  times.  He 
placed  his  finger 
on  his  pulse  and 
verified  the  truth 
of  his  first  impres- 
sion. Further  ex- 
periment showed 
that  the  time  of 
a  pendulum  de- 
pends upon  its 
length,  which 
must  be  measured 
from  its  centre  of 
mass  to  the  point 
of  support,  and 
with  the  force  of 
gravity  at  the 
place.  It  may  be  proved  mathematically  that  the  period, 
T,  of  a  pendulum  of  length  Z,  is 


FIG.  137. 


(30)    T=2*\/  — 
9 

where  *  =  3.1416  and  g  is  the  force  of  gravity.  The 
time  of  oscillation,  £,  which  is  the  time  between  two 
passages  of  the  ball  through  0,  is  just  half  as  great : 

(81)    t  =  8.1416  \J— 
A    pendulum    whose    time    of   oscillation,    £,  is    one 


196  VIBRATIONS.     WAVES. 

second  is  called  a  seconds  pendulum.     Its  length  may 
be  easily  found  from  the  equation  (31) 


t  =  3.1416 


\/JL 


=  (3.1416?  - 


I  = 

The  same  formula  will  give  us  the  value  of  g  when 
we  have  measured  t  and  Z. 


It  will  be  seen  that  the  only  variable  quantities  in 
this  formula  (31)  are  I  and  g.  It  follows  that  the 
mass,  size,  and  material  of  the  pendulum  have  no  effect 
on  its  time  of  oscillation.  It  is  to  be  remarked  that 
formula  (31)  is  true  only  on  the  supposition  that  the 
amplitude  of  vibration  is  small  as  compared  to  the 
length  of  the  pendulum. 

A  simple  pendulum  is  one  in  which  the  weight  of  the 
string  is  so  small  that  the  centre  of  mass  of  the  pen- 
dulum can  be  thought  of  as  identical  with  the  centre 
of  mass  of  the  bob,  which  must  be  small.  If  the  bob 
is  a  ball  or  disc  we  measure  I  from  the  centre  of  the 
bob  to  the  point  of  support. 

The  length  of  a  pendulum  is  changed  by  tempera- 
ture, and  various  plans  have  been  devised  for  correcting 
the  error  thus  caused.  Descriptions  of  compensating 


WAVE  MOTION.  197 

pendulums  may  be   found    in    the    larger    treatises    on 
physics  or  in  any  good  cyclopaedia. 

122.  Wave  Motion.  —  The  particles  of  solid  bodies 
are  so  related  to  each  other  that  if  any  particle  be  set 
in  vibration  it  will  impart  its  motion  to  the  particles 
nearest  to  it,  these  particles  in  turn  passing  the  motion 
to  their  neighbors.  Thus  the  slamming  of  a  heavy 
door  in  a  large  building  will  send  tremors  to  every  part 
of  the  building.  The  explosion  of  a  powder  magazine 
will  shake  buildings  miles  away. 

Fluids  also  transmit  disturbances,  not  by  being  dis- 
torted as  solids  are,  for  they  have  no  form,  but  by 
being  compressed. 

A  disturbance  which  is  transmitted  from  particle  to 
particle  through  a  medium  is  called  a  wave. 

There  are  but  two  ways  known  to  us  in  which  energy 
may  be  transmitted  from  body  to  body:  (1)  by  the 
motion  of  bodies  themselves,  as  when  a  bullet  is  fired 
or  a  stream  of  water  flows;  (2)  by  waves,  as  when 
pressure  is  imparted  to  the  water  in  closed  pipes  or  to 
the  ether  surrounding  the  sun,  and  the  pressure  is 
transmitted  to  all  points  in  any  way  connected  with  the 
source  of  energy. 

Every  vibrating  body  sends  waves  through  the  bodies 
it  touches.  Wave  motion  is  therefore  present  every- 
where. As  heat  and  sound  and  light  it  assails  us  on 
every  hand.  The  study  of  wave  motion  is,  therefore, 
of  the  highest  interest  to  us. 


198  VIBBA  TIONS.     WA  VES. 

We  have  all  noticed  that  a  pebble  dropped  in  still 
water  will  send  a  wave  to  every  part  of  the  pool.  The 
reason  is  obvious:  the  water  displaced  by  the  pebble 
heaped  upon  the  water  near  it  presses  upon  it,  falls,  and 
is  carried  as  far  below  the  level  surface  as  it  was  above 
it  first,  as  the  water  did  in  the  dish  shown  in  Fig.  135. 

The  pressure  is  transmitted  to  all  points  of  the  water 
until  in  time  the  wave  has  reached  every  point. 

When  the  water  at  any  point  as  P  (Fig.  138)  has 


FIG.  13«. 

made  a  vibration  and  a  fourth,  and  is  starting  up  for 
the  second  time,  there  are  a  number  of  particles  at 
equal  distances  from  P  which  are  just  starting  up  for 
the  first  time  (R,  R).  The  distance  PR  is  called  a 
wave  length,  and  all  particles  which  are  any  whole  num- 
ber of  wave  lengths  distant  from  P  are  said  to  be  in 
the  same  phase.  The  distance  PR  we  shall  call  I.  It 
is  evident  that  if  every  vibrating  particle  in  the  medium 
makes  n  vibrations  per  second,  the  rate  of  propagation 
of  the  disturbance,  or,  as  we  say,  the  velocity,  v,  of  the 

wave,  is 

(33)    v  =  nl 

This  expression  is  true  for  all  kinds  of  wave  motion. 
The  velocity  of  waves  of  any  sort  in  a  particular 
medium  is  constant,  but  when  the  waves  pass  into  a 
different  medium  the  velocity  will,  in  general,  change. 


DIRECTION  OF   WAVES. 


199 


123.  Direction  of  Waves.  —  A  wave  which  travels 
along  the  bounding  surface  between  two  fluids,  like  air 
and  water,  travels  outward  in  concentric  circles.  A 
wave  in  a  string  travels  along  the  string.  A  wave  in  a 
large  body  of  any  homogeneous  medium  travels  out- 
ward from  the  centre  of  disturbance  in  concentric 
spheres.  The  direction  of  vibration  of  the  particle 
may  be  the  same  as  the  direction  of  propagation  of  the 


2.OO  O  O  OOOOOOOOOOO  O  O  OO 
P        O         P         Of        T* 


FIG.  139. 

wave:  the  wave  motion  is  then  said  to  consist  of 
longitudinal  vibrations  ;  or  the  particles  may  vibrate  in  a 
direction  at  right  angles  to  the  direction  of  propagation 
of  the  wave  :  the  vibrations  are  then  said  to  be  trans- 
verse? The  waves  on  the  surface  of  water  and  the 
waves  which  are  sent  along  a  clothesline  when  it  is 
struck  with  a  stick  are  transverse.  The  waves  of  pres- 


200  VIBE  A  TIONS.     WA  VE8. 

sure  which  are  imparted  to  water  in  pipes  by  means  of 
a  force  pump  are  longitudinal. 

The  two  sorts  of  waves  may  be  represented  diagram- 
ma  tically  as  in  Fig.  139,  where  the  circles  represent 
particles  of  the  medium  in  various  phases  of  vibration. 
At  1  a  transverse  wave  is  shown,  at  2  a  longitudinal 
wave.  To  save  time  in  drawing  the  figure  it  is  cus- 
tomary to  represent  both  sorts  of  waves  as  at  3,  leaving 
the  sort  of  wave  to  be  stated  in  the  discussion.  The 
distance  PR  •=  I  is  a  wave  length,  and  P  and  R  are  in 
the  same  phase,  while  P  and  P1  or  0  and  0'  a-re  in 
opposite  phases. 

124.  Reflection.  —  It  was  remarked  above  that  waves 
change  their  velocity  on  entering  a  different  medium. 

It  is   to  be 
obs  erved 

FI0140  further  that 

the-    energy 
Of  a  train  of 
FIG- 141-  waves    in 

any  medium  is  never  all  transmitted  to  the  second 
medium;  a  part  of  it  is  always  reflected  or  turned 
back  at  the  bounding  surface.  If  the  velocity  of  the 
waves  is  less  in  the  second  medium  than  in  the  first, 
the  reflected  wave  will  be  opposite  in  phase  to  the  orig- 
inal wave  (see  Fig.  140).  If  the  velocity  is  greater  in 
the  second  medium  the  wave  will  be  reflected  without 
change  of  phase  (see  Fig.  141). 


ADDITION  OF  WAVES.  201 

Except  in  the  cases  of  large  bodies  of  water  and  the 
atmosphere  we  have  to  deal  with  waves  in  a  body  of 
limited  size,  and  must  therefore  have  to  do  with  waves 
at  bounding  surfaces.  It  will  be  well  for  us,  therefore, 
to  examine  the  subject  somewhat  more  in  detail. 

125.  Addition   of    Waves.     Stationary  Waves.  - 
When  two  waves  pass  any  portion  of  a  medium  at  the 
same  time,  each  wave  produces  the  same  effect  on  every 
particle  of  the  medium  as  if   it  alone  were 
passing.     This  is  in  accordance  with  Newton's 
second  law  of  motion.     The  resultant  motion 
is,  therefore,  the  geometrical  sum  of  the  two 
motions,  and  the  resultant  wave  may  be  quite 
different  from  either  of  the  two  components. 


If  one  end  of  a  rubber  tube  be  attached  to 


\ 


a  hook  in  the  ceiling  while  the  other  end  is 
fastened  to  a  hook  in  the  floor  or  table,  waves 
may  be  sent  up  toV  tube  by  striking  a  quick 
blow  near  the  lower  end  of  the  tube.  When 
a  wave  reaches  the  top  it  will  be  reflected 
V  s  with  a  change  of  phase.  If  at  the  instant 
FJG  the  reflected  wave  starts  downward  a  second 
142-  one  is  sent  up  from  8  (Fig.  142),  the  two  143' 
waves  will  meet  midway  and  interfere.  It  is  easy  to 
see  that  a  particle  halfway  up  (Fig.  143)  is  pulled  in 
opposite  directions  by  almost  equal  forces  and  will  there- 
fore remain  at  rest.  If  a  continuous  series  of  waves  be 
sent  from  S  at  equal  intervals  they  will  all  interfere 


202 


VIBRA  TION8.     WA  VES. 


so  as  to  produce  rest  at  the  middle  but  motion  in  the 
other  parts  of  the  tube,  which  will  appear  as  in  Fig. 
144.  The  point  at  rest  is  called  a  node,  while  the  points 

of  greatest  activity  are  called 

antinodes.     We  shall  indicate 

nodes  by  N,  antinodes  by  A 

hereafter.     By  timing  the 

waves  suitably  we  may  pro- 
duce two  nodes  besides  those 

at  the  end,  as  in  Fig.  145,  or 

three    or   four   or   more    at 

pleasure.     In  every  case  the 

distance  between  two  nodes 

is  half  a  wave  length. 

(34)     NN=AA=L 

The  case   just  cited  is    an 
FIG.    FIG. 
144.     145.     illustration  of  the  general 

case  shown  in  Fig.  140,  where  vl  >  v2. 
The  case  where  vl  <  v2  may  be  illustrated  if  we  fasten 
the  upper  end  of  our  tube  to  a  string  which  is  fastened 
to  the  hook  as  shown  in  Fig.  146.  The  reflected  and 
incident  waves  (as  waves  sent  from  S  are  called)  will 
evidently  interfere  in  such  a  way  as  to  reenforce  each 
other  as  they  pass ;  the  middle  portion  will  not  be  a 
node,  neither  will  the  upper  end,  but  the  nodes  will  be 
situated  as  shown  in  Figs.  147  and  148. 


FIG.   Fio.     FIG. 
146.      147.        148. 


126.  Law  of  Reflection.  —  The  wave  in  a  rope  or 


LAW  OF  REFLECTION. 


203 


rubber  tube  is  reflected  back  along  the  tube  as  a  matter 
of  course,  but  when  a  water  wave,  for  example,  strikes 
a  pier  the  direction  of  the  reflected  wave  is  not  exactly 
opposite  to  that  of  the  incident  wave  unless  the  latter 
was  moving  at  right  angles  (normal)  to  the  pier.  Let 
fit  f'i  (Fig-  149)  be  the  Avave  fronts  of  a  train  of  waves 
incident  upon  the  bounding  surface  BB',  fr,  ffr  the 
front  of  the  same  wave  after  reflection.  The  waves  are 
supposed  to  have  come  from  a  point  so  distant  that  the 


FIG.  149. 

wave  fronts  may  be  treated  as  straight  lines.  When  a 
certain  point  C  on  f{  has  reached  the  bounding  surface 
BB',  there  is  a  point  C1  on  f\  which  has  also  just 
reached  BB'.  If  we  draw  CD  _L  ft  and  C'I>'  J_  /r, 
it  is  evident  that  D  will  reach  C  in  the  same  time 
that  C'  requires  to  reach  D1,  whence  D  C  =  D'  C'  =  I. 
In  the  right-angled  triangles  C'DO  and  CD'C'  the  sides 
C'D*  and  CD  are  equal  and  C'C  is  common.  They  are 
therefore  equal  and  angle  DC'C  =  angle  D'OO' ;  that  is 


204 


vis  MA  r/o.v/y.    WA  VES. 


to  say :    the  angles  made  by  the  incident  and  reflected 
waves  with  the  bounding  surface  are  equal. 

127.  Refraction.  — In  Fig.  150  /,,  /',  are  the  fronts 
of  waves  incident  on  BB'  and  fR,  f'R  the  fronts  of  that 
portion  of  the  waves  which  enter  the  second  medium. 
When  f{  has  just  touched  BB'  at  (7,  /',.  has  also  just 
touched  BB'  at  (7;,  while  other  portions  of  these  waves, 


FIG.  150. 


/j,  f'x,  have  advanced  at  a  different  velocity  in  the  sec- 
ond medium.  Draw  CD'  J_  /',  and  C'D  _L  fR.  The 
distance  D' C  was  traversed  hy  the  wave  in  the  first 
medium  in  the  same  time  that  the  distance  C'D  was 
traversed  in  the  second  medium,  or 
D'C 


C'D 


-  =  n,  a  constant 


The  velocity  of  any  particular  waves  in  any  given 
medium  is  constant  and  the  ratio  v1/v2  is  therefore  con- 
stant. It  is  known  as  the  index  of  refraction  and  is 


EXERCISES.  205 

denoted  by  n.  Refraction  may  be  defined  as  the 
change  of  direction  which  a  wave  undergoes  in  passing 
from  one  medium  to  another.  It  is  evident  from  the 
figure  that  the  direction  of  the  wave  front  will  be 
changed  except  when  the  wave  front  is  parallel  to  the 
bounding  surface,  that  is,  when  the  angle  between  the 
incident  wave  and  the  bounding  surface  is  zero.  In 
general,  the  greater  the  angle,  the  more  the  wave  will 
be  refracted. 

It  should  be  remarked  that  reflection  and  refraction 
take  place  in  the  manner  just  described  only  when  the 
bounding  surfaces  are  plane  over  an  area  which  is  large 
compared  with  the  length  of  a  wave. 

The  subject  of  diffraction  may  well  be  deferred  until 
light  waves  are  studied. 

Exercises. 

102.  («)  What  is  the  length  of  the  seconds  pendulum  at  a 
place  where   #=979?     (£)  What   would   be   the  rate  of  this 
pendulum  at  a  place  where  ^\=  981  ? 

103.  At  a  certain  place  a,  pendulum  was  found  to  make  21 
oscillations  in  20  seconds.     Its  length  was  90.2  cm.     What  is 
the  value  of  g  at  that  place  ? 

104.  Two  pendulums  have  lengths  in  the  ratio  of  3  to  4. 
What  is  the  ratio  of  their  periods  ? 

105.  Place  a  rectangular  dish  of  mercury  or  water  on  the  table 
and  strike  a  series  of  blows  on  the  table  with  the  fist.     Strike 
simultaneously  with  both  fists  at  points  at  right  angles  to  two 
adjacent  sides  of  the  dish  and  note  the  waves  on  the  surface  of 
the  liquid.     A  whirling  table  or  an  electric  motor  in  motion 
will  often  set  up  such  waves, 


206 


VIBE  A  TIONS.     WA  VE8. 


106.  A  rope  along  which  impulses  are  being  sent  at  the  rate 
of  5  per  second  vibrates  in  4  segments.     The  rope  is  6  metres 
long,     (a)  What  is  the  wave  length  ?    (6)  What  is  the  velocity 
of  the  wave  ? 

107.  Plot  on  cross-section  paper  a  train  of  waves  having  a 
wave  length  of  4  cm.  and  an  amplitude  of  1  cm.,  and  a  second 


FIG.  161. 

train  having  a  wave  length  of  3  cm.  and  an  amplitude  of  1  cm. 
Plot  the  sum  of  the  two  waves  by  taking  the  sum  or  difference 
of  the  distances  from  the  axis  of  points  2  mm.  apart  along  the 
axis  as  illustrated  in  Fig.  151,  where  the  lengths  are  respec- 
tively 1  and  2. 


CHAPTER   VII. 
SOUND. 

128.  Nature  of   Sound.  —  Sound  is  a  sensation  re- 
ceived by  the  auditory  nerve.     It  has  its  origin  in  some 
vibrating  body,  as  we  may  easily  convince  ourselves  by 
observation.     If  we  touch  a  sounding  string  or  bell  we 
perceive  that  it  is  in  a  state  of  vibration.     Not  only  so, 
but  when  we  stop  its  vibrations  by  touching  it  the  sound 
ceases.     The  vibrations  of  the  sounding  body  are  con- 
veyed to  the  -ear  by  waves  in  an  elastic  medium,  usually 
the  air,  which  sets  in  vibration  the  drum   of  the  ear. 
The  way  in  which  the  vibrations  of  the  ear  drum,  after 
being  transmitted  by  the  chain  of  bones  in  the  middle 
ear  to  the  fluids  of  the  inner  ear,  are  analyzed  and  per- 
ceived as  separate  impressions  is  not  well  understood. 
In  physics  we  are  more  directly  concerned  with  the  con- 
ditions requisite  for  producing  and  transmitting  to  the 
ear  the  vibrations.     These  are  pouring  in  upon  us  from 
every  side  even  in  our  hours  of  sleep,  and  we  cannot 
shut  them  out  as  we  shut  out  light.     Indeed,  the  pre- 
vention of  the  vibrations  which  produce   sound  must 
one  day  become  a  very  important  factor  in  the  comfort 
and  health  of  people  who  live  in  cities. 

129.  Noises  and  Musical  Notes.  — The  ear  delights 

207 


208  SOUND. 

in  order.  A  rapid  succession  of  waves  of  the  same  sort 
produces  a  pleasant  impression  upon  the  ear,  while  a 
confused  mingling  of  many  sorts  of  waves  produces 
a  sound  which  is  disagreeable.  The  confused  rattle  of 
a  moving  railway  train,  the  squeaking  of  the  wheels 
against  the  track  as  a  curve  is  rounded,  and  the  flapping 
of  the  brake  shoes  against  the  wheels  are  not  pleasing 
to  the  ear  of  the  passenger,  but  when  the  train  stops  at 
the  end  of  a  division  and  the  wheel  tester  strikes  each 
wheel  in  turn  with  his  hammer  we  notice  that  each 
wheel  gives  a  clear  musical  note.  We  shall  find,  if  we 
test  various  bodies  like  sticks,  tables,  dishes,  stones, 
iron  bolts,  indeed  any  hard  body  which  is  free  from 
loose  parts,  that  each  body  has  a  definite  note  of  its 
own. 

130.  Vibrating  Rods.  —  If  we  select  two  wooden 
rods  of  equal  dimensions  we  shall  find  that  they  give 
notes  nearly  alike.  By  sawing  one  into  two  unequal 


FIG.  152. 


parts  we  shall  get  two  rods  which  will  give  notes  which 
are  not  like  either  the  long  rod  or  each  other.  Let  us 
see  why  there  should  be  a  difference  between  them. 
Let  Fig.  152  represent  three  wooden  rods  resting  on 


BELLS  AND  PLATES.  209 

bits  of  rubber  tubing.  They  are  alike  in  all  respects 
save  length.  If  a  blow  be  struck  at  S  a  wave  will 
travel  to  FJ  where  it  will  be  reflected  and  return  to  S  to 
be  again  reflected.  A  series  of  waves  will  be 
sent  out  into  the  air  at  intervals,  which  depend 
upon  the  time  taken  for  the  wave  to  traverse 
the  stick.  The  ends  of  the  stick  are  free  and 
are  consequently  antinodes.  There  will  be 
another  antinode  at  the  centre,  with  nodes  at 
points  one  fourth  of  the  length  of  the  stick 
from  each  end. 

The  wave  length  of  the  stationary  wave  is 
the  same  as  the  length  of  the  rod.  Each  part 
of  the  rod  vibrates  like  a  pendulum,  and  for  rods 
which  are  alike  in  all  other  respects  the  period 
of  vibration,  like  that  of  the  pendulum,  varies 
with  the  square  of  the  length.  A  thick  rod 
vibrates  more  slowly  than  a  thin  one  of  the 
same  length.  Metal  rods  emit  notes  which 
are  purer  and  louder  than  those  given  by 
wooden  rods.  The  tuning  fork  (Fig.  153) 
is  a  steel  rod  bent  double  and  provided  with 
a  handle  at  its  middle  point.  The  bending  FlG- 154> 
of  the  fork  brings  the  nodes  nearer  together  than  they 
would  otherwise  be,  and  the  fork  vibrates  as  indicated 
in  Fig.  154. 

131.  Bells  and  Plates.  —  The  vibrations  of  plates 
are  analogous  to  those  of  rods,  while  the  vibrations  of 


T 


210 


SOUND. 


bells  are  somewhat   similar  to   those  of  forks.     When 
a  bell  is  struck  as  at  A  (Fig.  155),  its  form  is  altered 

from  the  circular  £0 
a  slightly  elliptical 
shape.  Its  elasticity 
draws  it  back  and 
carries  it  past,  as  in 
other  vibrating  bod- 
ies. The  result  is  to 
set  the  bell  vibrating 
in  four  segments. 
This  may  be  beauti- 
fully shown  by  filling 
a  jar  or  glass  half  full 
of  water  and  striking 
the  rim.  The  parts 
next  the  antinodes  will  be  thrown  into  ripples,  while  the 
parts  next  the  nodes  remain  comparatively  quiet.  If 
the  bell  is  struck  at  a  point  which  is  slightly  heavier 
than  some  point  near  it,  this  point  will  tend  to  become 
a  node 'and  there  will  be  a  curious  changing  effect 
known- as  beats.  This  will  be  explained  in  Section  134. 

132.  Pitch. — When  two  forks  of  different  dimen- 
sions are  struck  in  succession  we  have  no  difficulty  in  i 
deciding  that  the  tones  they  give  are  different.  If  one 
is  much  larger  than  the  other  we  shall  all  agree  that  the 
larger  gives  a  tone  which  differs  from  that  of  the  small 
one,  just  as  the  tones  of  a  man  differ  from  those  of  a 


FIG.  155. 


MUSICAL  INTEEVALS.  211 

child ;  the  long  fork  has  a  lower  pitch,  as  we  say.  The 
pitch  of  a  note  is  determined  by  its  vibration  frequency, 
n.  The  pitch  of  women's  voices  is,  on  the  average, 
twice  as  high  as  that  of  men.  The  pitch  of  a  rod  20 
cm.  long  is  four  times  as  high  as  that  of  a  rod  of  the 
same  thickness  but  40  cm.  long.  A  tuning  fork  which 
is  not  exactly  of  the  proper  pitch  may  be  tuned  by  fil- 
ing off  the  ends  if  it  is  too  low  or  by  filing  it  thinner 
if  it  is  too  higlv  The  variation  of  pitch  with  frequency 
may  be  easily  shown  by  holding  a  card  against  a  toothed 
wheel  which  is  made  to  revolve  at  different  speeds,  or 
by  directing  a  blast  of  air  from  a  tube  which  has  a  small 
nozzle  toward  a  row  of  holes  in  a  revolving  disc  of 
cardboard  or  metal.  Such  a  disc,  called  a  siren,  is  usu- 
ally provided  with  several  rows  of  holes,  so  that  the 
pitch  may  be  changed  by  passing  the  nozzle  from  one 
row  of  holes  to  another  without  changing  the  speed  of 
the  disc.  If  we  let  the  speed  of  the  toothed  wheel  or 
siren  fall  till  the  frequency  is  less  than  about  sixteen 
vibrations  per  second,  we  shall  begin  to  hear  the  separate 
impulses.  When  the  number  of  vibrations  exceeds 
40,000  per  second  we  cease  to  be  able  to  hear  them  at 
all.  It  is  quite  probable  that  certain  insects  hear  notes 
which  our  ears  are  quite  unable  to  perceive  because  of 
their  high  pitch. 

133.  Musical  Intervals.    Consonance.    Dissonance. 

-When  two  musical  notes   are  sounded  at  once  the 
effect  upon  the  ear  is  sometimes  pleasant  and  sometimes 


212  SOUND. 

very  disagreeable.  It  has  been  found  that  this  difference 
depends  on  the  relative  vibration  frequency  of  the  two 
notes.  The  ratio  of  the  frequency  of  two  notes  is 
called  the  musical  interval  between  them.  If  this  inter- 
val can  be  expressed  by  small  numbers  the  combined 
effect  is  pleasant  to  the  ear,  and  the  notes  are  said  to  be 
consonant.  Intervals  which  can  be  expressed  only  by 
large  numbers  are  dissonant.  Primitive  music  concerned 
itself  mainly  with  the  order  of  succession  of  the  differ- 
ent notes,  the  same  tune  being  followed  by  all  voices ; 
modern  music  combines  several  tunes  or  parts  in  one 
harmony,  the  variety  and  richness  of  which  is  still  fur- 
ther enhanced  in  the  orchestra  by  employing  instru- 
ments which  give  tones  of  different  quality.  We  shall 
recur  to  the  matter  of  quality  in  a  later  section,  but  it 
will  be  found  to  be,  after  all,  dependent  upon  frequency. 
Vibrations  differ  only  in  two  particulars  —  frequency  and 
amplitude^  The  amplitude  of  vibration  determines  the 
loudness  of  the  sound.  The  remaining  characteristics 
of  a  musical  note,  namely,  pitch  and  quality,  are  depend- 
ent upon  the  rate  of  vibration. 

134.  Beats. — If  two  tuning  jforks  have  the  same 
pitch  they  will,  when  sounded  together,  produce  the 
most  perfect  consonance.  The  interval  is  i.  If  now 
we  attach  a  bit  of  wax  to  a  prong  of  one  of  the  forks 
its  rate  will  be  slightly  diminished ;  the  waves  from  the 
two  forks  when  sounded  together  will  interfere,  first  re- 
enforcing,  then  destroying  each  other.  The  effect  pro- 


OVERTONES   OB  PARTIAL  S.  213 

duced  is  called  a  beat.  It  is  disagreeable,  producing 
much  the  same  effect  upon  the  ear  that  a  flickering 
light  does  upon  the  eye.  The  number  of  beats  per  sec- 
ond is  equal  to  the  difference  between  the  vibration 
numbers  of  the  two  notes.  Tuners  of  musical  instru- 
ments make  use  of  beats  in  tuning  the  instrument. 
They  have  only  to  seek  to  diminish  the  number  of 
beats  until  beats  are  no  longer  heard ;  when  the  string 
or  pipe  is  known  to  be  in  unison  with  the  standard  fork. 
Beats  in  bells  are  overcome  to  some  extent  by  chipping 
away  with  a  chisel  the  parts  which  are  too  heavy. 

135.  Overtones  or  Partials.  —  We  saw  in  Section 
125  that  a  body  of  limited  dimensions,  like  a  string 
with  its  ends  fastened,  may  vibrate  as  a  whole  or  in  any 
whole  number  of  equal  parts.  It  has  been  found  that 
most  vibrating  bodies  when  vibrating  as  a  whole  also 
vibrate  in  segments  at  the  same  tii^e.  When  a  body 
vibrates  as  a  whole  it  gives  the  lowest  note  which  it  is 
capable  of  giving.  This  is  called  its  fundamental  tone: " 
The  highest  tones  which  it  gives  when  vibrating  in 
parts  are  called  partiah  or  overtones.  When  it  vibrates 
in  two  parts  it  gives  its  first  overtone,  in  three  parts  its 
second  overtone,  and  so  on.  The  first  five  partial  tones 
are  perfectly  consonant  with  the  fundamental  and  each 
other,  for  their  intervals  are  all  expressed  in  small  num- 
bers :  f,  f ,  f ,  |,  A  4,  etc. 

The  overtones  given  by  any  particular  sounding  body 
will  depend  much  upon  the  shape  and  material  of  the 


214  SOUND. 

body  itself.  Since  the  overtones  present  in  one  voice  or 
instrument  are  different  from  those  in  another,  the  tone 
produced  by  their  combination  with  the  fundamental  will 
be  different.  The  pitch  of  the  note  we  judge  from  the 
pitch  of  the  fundamental,  which,  if  present  at  all,  is  usu- 
ally the  prevailing  note.  The  minor  differences  produced 
by  the  peculiarities  of  the  instrument  we  call  the  quality 
of  the  note.  Strings  break  very  readily  into  segments 
and  are  therefore  rich  in  overtones.  A  board  can  vi- 
brate in  almost  any  number  of  parts,  yet  has  no  marked 
preference  for  its  fundamental.  Boards  are  therefore 
well  adapted  to  take  up  the  vibrations  of  strings  and 
forks  which  are  in  contact  with  them,  thus  reenforcing 
the  note  of  the  string  or  fork. 

136.  Resonance.  Pipes.  —  If  one  lifts  the  dampers 
from  all  the  strings  of  a  piano  and  sings  a  single  note, 
the  strings  corresponding  to  the  fundamental  and  the 
principal  overtones  of  the  note  sung  will  be  set  in  vi- 
bration and  may  be  distinctly  heard.  A  window  in  a 
church  will  often  be  set  rattling  when  a  particular  note 
of  the  organ  is  sounded.  This  phenomenon  is  called 
resonance.  It  is  due  to  the  fact  that  the  resounding 
body  has  the  same  natural  period  of  vibration  as  the 
sounding  body,  so  that  the  impulses  given  by  the  sound- 
ing body  are  timed  just  right  to  reenforce  each  other. 
The  reenforcement  of  a  tuning  fork  by  a  column  of  air 
is  easily  shown  by  holding  a  fork  which  has  just  been 
struck  over  a  tall  jar  and  slowly  pouring  water  into  the 


RESONANCE.     PIPES. 


215 


A 


jar  to  shorten  the  air  column.  When  the  air  column  is 
exactly  the  right  length  the  fork  will  speak  loudly 
enough  to  be  heard  distinctly  all  over  the  room  (see 
Fig.  156).  The  bottom  of  the  pipe  will  be  a  node  and^ 
the  top  will  be  an  antinode,  so  that  the  length  of  the 
column  would  be  exactly  1 1  if.  it  were  not  for  the  fact 
f,  that  the  size  and  shape 

of   the   opening    at   the 

top  changes  slightly  the 

position  of  the  antinode, 

throwing  it  a  little  dis- 
tance above   the  top  of 

the.  jar.     If  a  jar  three 

times  as  tall  as  the  short- 
est column  of  air  which 

will   reenforce  the   fork 

is    used    resonance  will 

again  occur,  for  the  air 

column  will  break  into 

segments,  as  shown  in 
FIG.  156.  Fig.  15T.  The  reso- 
nance of  air  columns  is  employed  in  organ  pipes,  the 
pipes  being  made  of  the  proper  length  to  reenforce  the 
different  notes.  The  air  is  set  into  irregular  vibrations 
by  being  blown  against  the  mouthpiece  in  Fig.  158. 
Those  vibrations  which  are  of  the  right  period  to  be 
reenf  orced  by*the  pipe  are  the  ones  which  we  hear.  The 
closed  pipe,  like  the  resonance  jar,  is  one  fourth  the 
length  of  a  wave  in  air.-  The  open  pipe  (Fig.  159)  has 


FIG.  157. 


216  SOUND. 

an  antinode  at  each  end.     Its  length,  therefore,  is  half 
a  wave  length,  hence  it  must  be  twice  as  long  as  a  closed 
pipe  which  is  to  give  the  same 
note. 

Organ  pipes  are  often  pro- 
vided with  reeds  like  the  reeds 
of  an  accordion  or  harmonica 
(mouth  organ).  Organ  pipes 
are  often  lacking  in  some  of  the 
higher  overtones.  The  lack  is 
made  good  by  smaller  pipes 
which  are  so  connected  to  the 
keys  as  always  to  speak  when 
the  fundamental  does. 

Such  wind  instruments  as  the 

flute  and  cornet  have  means  of         FIG* 159> 
FIG.  158. 

varying  the  length  of  the  air  column  to  pro- 
duce a  number  of  different  notes.  Each  of  the  funda- 
mental lengths  can  be  made  to  serve  for  a  number  of 
notes  by  varying  the  tension  of  the  lips  and  the  pressure 
of  the  breath,  as  is  done  in  the  bugle.  The  fundamental 
note  of  such  instruments  is  not  used. 

137.  Velocity  of  Sound  Waves. — The  resonance 
jar  furnishes  a  convenient  method  of  measuring  the 
velocity  of  sound  in  air.  Thus  in  Fig.  160 

(34)     NN=  I  I 
v  =  nl 


whence :     (36)         v  =  2n  NN 


VELOCITY  OF  SOUND    WAVES. 


217 


where  the  vibration  number  of  the  fork  is  supposed  to 
be  known.  The  velocity  of  sound  varies  inversely  with 
the  square  root  of  the  density  of  the 
medium.  Since,  therefore,  air  expands 
by  heating,  sound  travels  faster  in 
warm  air  than  in  cold.  The  velocity  of 
sound  in  air  at  0°  C.  is  332  metres  per 
second.  At  any  temperature,  t,  it  is 


FIG.  160. 


(37)    v  =  332  t/  1  +  .004  t 

since  air  expands  ^-^  =  .004  for  each 
degree  Centigrade  above  0°  C. 

The  velocity  of  sound  in  solids  may 
be  found  by  a  beautiful  method  due  to 
Professor  Kundt.  He  clamps  a  rod  at 
its  middle  point  and  attaches  a  cork 
to  one  end,  which  is  allowed  to  project 
into  a  long  glass  tube  (see  Fig.  161).  The  bottom  of 
the  tube  as  it  lies  in  a  horizontal  position  is  covered  with 
fine  cork  filings,  and  the  other  end  of  the  tube  is  closed 
with  a  movable  cork  which  fits  snugly  and  is  used  to 


FIG.  lei. 

vary  the  length  of  the  tube.  The  rod  is  stroked  with 
a  rosined  cloth  and  thus  set  into  rapid  longitudinal  vibra- 
tions. The  cork,  which  fits  loosely  in  the  tube,  sets  the 
air  in  the  tube  in  vibrations,  which  by  reflection  from 
the  farther  end  produce  stationary  waves  in  the  tube. 


218  SOUND. 

The  dust  at  the  antinodes  is  violently  agitated  and  falls 
into  little  ridges  when  the  sound  ceases.  The  distance 
between  the  antinodes  is  easily  measured.  The  middle 
of  the  rod  is  a  node  and  the  ends  antinodes.  The  length 
of  the  rod  is,  therefore,  half  the  length  of  a  wave  in  the 
solid.  If  va  is  the  velocity  of  sound  waves  in  air  at 
the  given  temperature,  then  vs,  the  velocity  in  the  solid 
used,  is 


where  ls ,  la  are  the  lengths  of  a  wave  in  the  solid  and 
in  air  respectively. 

It  should  be  borne  in  mind  that  all  stationary  waves 
are  caused  by  the  interference  of  waves  reflected  from 


FIG.  162. 

opposite  bounding  surfaces.  The  length  of  a  wave 
may  be  measured,  also,  by  the  interference  of  two  waves 
from  the  same  point,  which  travel  by  different  paths  to 
a  second  point,  as  shown  in  Fig.  162.  If  one  branch 
of  the  rubber  tube  is  half  a  wave  length  longer  than 
the  other  for  waves  of  the  frequency  of  the  fork 
which  is  vibrating  at  S,  the  waves  will  meet  at  E  in 


VIBRATING   STRINGS.  219 

opposite   phase  and   produce    silence.     An   interesting 
example  of  the  production  of  silence  by  the  interfer- 
ence of  two  sound  waves  is  the  case  of  a  tuning  fork. 
If  a  vibrating  tuning  fork  be  slowly  rotated  near  the 
ear  or  over  a  resonance  jar,  four  positions  may  be  found 
where  no  sound  is  heard. 
If  while   the  fork  is  in 
one  of  these  positions  a 
paper   tube   be    slipped 
over  one  prong  of   the     R  ®  */i\ 

fork  so  as  not  to  touch 
it,  the  sound  will  again 
be  heard.  This  is  read-  , 

rIG.  loo. 

ily  understood  by  refer- 
ring to  Fig.  163.  When  the  prongs  approach  each  other 
condensations  are  sent  toward  (7,  rarefactions  toward  R. 
At  all  points  on  the  diagonal  lines,  which  are  equally 
distant  from  0  and  R,  the  waves  will  meet  in  opposite 
phases  and  destroy  each  other. 

138.  Vibrating  Strings.  —  We  have  referred  so 
often  to  the  vibrations  of  stretched  strings  that  the 
nature  of  these  vibrations  should  now  be  pretty  well 
understood.  The  mathematical  laws  governing  the 
vibration  frequency  of  strings  are  easily  demonstrated 
by  experiment.  They  may  be  stated  as  follows : 

The  number  of  vibrations  per  second  of  a  tightly 
stretched  string  which  is  giving  its  fundamental  note 
is : 


220  SOUND. 

1.  Inversely  proportional  to  its  length. 

2.  Directly  proportional  to  the    square    root  of  the 
tension.  Vy — 

3.  Inversely  proportional  to  its  linear  density. 
These  three  laws  may  be  summed  up  in  one  formula : 


where  L  is  the  length  of  the  string,  /  is  the  tension  in 
dynes,  and  m  the  mass  per  unit  length  of  the  strings. 

These  laws  are  all  exemplified  by  such  stringed  in- 
struments as  the  guitar  and  the  violin.  The  player 
raises  the  pitch  of  any  string  by  touching  it  so  as  to 
shorten  the  vibrating  portion.  He  raises  the  pitch  in 
tuning  by  increasing  the  tension.  The  heavy  strings 
are  the  ones  which  give  the  lower  notes. 

The  point  at  which  a  string  is  struck  or  bowed  affects 
its  quality  by  determining  to  some  extent  what  over- 
tones shall  be  present.  The  hammers  of  a  piano  are  so 
placed  as  to  prevent  the  occurrence  of  a  certain  higher 
overtone  which  is  not  consonant  with  the  others.  Over- 
tones are  often  spoken  of  as  harmonics.  This  term 
should  be  applied  only  to  such  overtones  or  partials  as 
are  consonant  with  the  fundamental  tone. 

139.  Musical  Scales.  —  The  foundation  of  all  our 
music  is  the  scale  of  eight  notes,  the  eighth  or  octave  of 
which  has  just  double  the  frequency  of  the  first.  The 
relative  frequencies  of  the  major  scale  are  variously 


MUSICAL   SCALES.  221 

represented  in  lines  1,  2,  and  3,  the  absolute  frequencies 
in  lines  4,  5,  and  6. 

1.  %        %         %          %         %         %       15/8        % 

2.  First  Second  Third  Fourth  Fifth  Sixth  Seventh  Octave 

3.  Do       Ke        Mi        Fa        Sol        La         Si        Do 


F5? 

—  /<3  —  • 

EBE:: 

^-. 

—  £2  — 

<S>  

£r  ^. 

P 

c' 

d' 

e' 

f 

o-' 

o 

a' 

b' 

c" 

264 

297 

330 

352 

396 

440 

495 

528 

5. 

6. 

This  scale  is  made  up  by  combining  three  major  triads, 
and  is  therefore  known  as  the  major  scale.  The  major 
triad  is  so  called  because  it  is  composed  of  three  notes 
which,  when  sounded  together,  give  the  most  perfect 
harmony,  namely,  the  first,  third,  and  fifth,  or,  Do,  Mi, 
Sol.  If  we  add  the  octave  it  gives  us  the  major  chord. 
Helmholtz  pointed  out  that  the  reason  for  the  perfect 
consonance  of  the  major  triad  lies  in  the  fact  that  no 
beats  occur  between  any  of  the  overtones  of  any  tone 
either  with  the  fundamentals  or  overtones  of  any  other 
of  the  triad.  This  may  be  illustrated  by  an  example. 

FIRST  THIRD  FIFTH 

Fundamental  200  250  300 

First  partial  400  500  600 

Second    „  600  750  900 

Third      „  800  1,000  1,200 

Fourth    „  1,000  1,250  1,500 

Fifth       „  1,200  1,500  1,800 


222  SOUND. 

It  will  be  seen  that  there  is  unison  between  the  second 
partial  of  the  first  and  the  first  partial  of  the  fifth,  as 
also  between  the  fourth  partial  of  the  first  and  the  third 
partial  of  the  third.  Indeed,  there  are  four  pairs  in 
unison.  But  in  no  case  is  the  difference  between  the 
number  of  vibrations  of  any  pair  less  than  the  difference 
between  the  fundamentals  of  the  first  and  fifth.  By 
way  of  contrast  let  us  compare  the  second,  fourth,  and 
sixth. 

SECOND  FOURTH  SIXTH 

Fundamental  225  267  333 

First   partial  450  534  666 

Second   „  675  801  999 

Third      „  900  1,168  1,332 

Fourth    „  1,125  1,335  1,665 

Fifth      „  1,350  1,582  1,998 

The  combination  666  and  675  is  bad,  but  when  we 
combine  1,350,  1,335,  and  1,332  the  effect  is  wholly 
unpleasant. 

The  relative  frequencies  of  the  major  triad,  as  will  be 
seen  from  the  table,  are : 

Vl  :  %  :  % 
c'  :  e'  :  g' 

or,  expressed  in  whole  numbers  : 

4:5:6 
c'  :   e'  :  g' 

If  we  form  a  second  triad  taking  c"  as  6,  and  still  an- 
other taking  g'  as  4,  we  have : 


'  MUSICAL   UCALES.  223 

4:5:6 
c'  :  e'  :  g' 

f   :   a'  :  c" 
g'  :  V  :  d" 

From  which  we  obtain  : 

f  :  c"  =  f  :  2c  =  4  :  6  whence  f  =  %  c' 
a' :  c"  =  a' :  2c  =  5  :  6  whence  a'  =  %  c 
g' :  b'  =4:5  and  g'  :  c'  =  6  :  4  .-.  b'  =  1%  c' 

The  major  scale  is  thus  seen  to  be  composed  of  three 
major  triads. 

It  is  readily  seen  on  inspecting  the  table  of  intervals 
below  that  there  are  five  intervals  of  nearly  equal  mag- 
nitude and  two  which    are   only  about  half   as  large. 
The  former  are  called  full  tones,  the  latter  semitones. 
Note,  c'       d'       e'        f        g'       a'        b'       c" 

•iveiative  -,  ,       ^ ,       ~/        */       o /       K /       -«  K /      r>  / 

frequency,      %      %      %       %      %      %       '%     % 

Intervals,  %     10/9    16/15     %     10/9     %      16/15 

To  make  possible  the  writing  of  music  in  scales 
higher  than  the  one  just  shown  (which  is  known  from 
its  keynote  as  c  major)  semitones  have  been  inserted 
between  c'  and  d'  and  in  the  other  intervals,  which  are 
full  tones,  by  sharping  or  flatting  the  notes  —  that  is,  by 
raising  them  ^  or  by  lowering  them  ^V  Thus:  c# 
(c  sharp)  is  equal  to  ||  c,  while  d!7  (d  flat)  is  |-|  d. 
The  semitone  added  is  not  halfway  between  c  and  d, 
and  ctf  and  d'p  are  not  identical,  for  c#  =  JJ  c  = 
|  .  264  =  275,  while  dfr  =  d  =  •  297  =  285. 


224  SOUND. 

In  ascending  the  scale  sharps  are  used,  in  descending 
flats  are  used.  The  scale  of  eighteen  notes  thus  formed 
is  called  the  chromatic  scale. 

The  tempered  scale  was  devised  for  use  with  instru- 
ments like  the  piano  and  organ,  which,  unlike  the  violin 
and  the  voice,  must  have  a  separate  string  or  pipe  of 
fixed  length  for  each  note.  To  avoid  the  excessive 
number  of  notes  required  the  octave  is  divided  into 
twelve  equal  intervals,  each  of  which  has  a  frequency 

12  /— 

equal  to  y  2  =1.059  times  the  note  below.  This 
arrangement  necessitates  throwing  all  intervals  except 
the  octave  slightly  out  of  tune.  Long  use  makes  us 
accustomed  to  the  lack  of  perfect  harmony,  so  that  we 
give  little  heed  to  it.  The  human  voice  can  hardly  be 
at  its  best,  though,  when  accompanied  by  the  piano. 

The  progress  of  music  as  an  art  side  by  side  with  the 
science  of  music  is  a  good  example  of  the  interdepend- 
ence of  all  human  efforts. 

Exercises. 

108.  (a)  What  length   of  closed   organ   pipe  will  give  c'? 
(6)  What  note  will  an  open  pipe  of  the  same  length  give  ? 

109.  (a)  Compute  the  linear  density  of  a  string  of  a  mono- 
chord    (Fig.    164)    after   measuring   its   diameter.     Make   the 
other  measurements  necessary  for  computing  n  in  formula  (39). 
(6)  Calculate  n  by  means  of  a  resonance  tube  made  of  a  large 
glass  tube  with  a  piston  for  varying  its  length,  and  see  how  the 
two  values  agree. 

no.  What  is  the  velocity  of  sound  waves  in  a  brass  rod  120 
cm.  long  which  excites  waves  24  cm.  long  in  a  Kundt's  tube? 
The  temperature  of  the  air  was  20°  C- 


EXERCISES.  225 


in.  (a)  Will  a  tuning  fork  vibrate  longer  in  tne  hand  or 
resting  on  a  table?  (6)  In  which  place  will  it  give  the  louder 
sound  ?  Explain. 

112.  An  echo  is  sent  back  (by  reflection)  from  a  barn.     The 
last  of  a  series  of  words  requiring  five  seconds  to  utter  can  be 
spoken  just  before 

the   first  word    is          9 ^ 

returned.     The         HSj 

air  is  at  the  tem- 
perature of  freez- 
ing water.  How 
far  away  is  the 
barn?  FlG' 164' 

113.  Two  telegraph  sounders  are  on  the  same  circuit,  which 
is  closed  twice  a  second  by  a  pendulum.     One  sounder  is  by 
the  observer's  side,  the  other  is  550  feet  away.     What  is  the 
velocity  of  sound  if  the  two  sounders  seem  to  tick  in  unison  ? 

114.  A  stone  is  dropped  down  a  well.     After  two  seconds 
the  splash  of  the  stone  in  the  water  is  heard.     How  far  is  the 
surface  of  the  water  below  the  top  of  the  well  ? 

115.  (a)  An  organ  pipe  gives  a  note  of  frequency  264  at 
16°  C.     What  will  it  give  at  28°  C.?     (6)  Would  a  piano  string 
which  is  in  tune  with  the  pipe  at  15°  be  in  tune  at  28°  C.? 

116.  A  tuning  fork  placed  at  S  (Fig.  162)  gives  the  mini- 
mum sound  at  E  when  the  long  arm  of  the  tube  is  65  cm. 
longer  than  the  short  arm.     The  temperature  is  20°  C.     What 
is  the  pitch  of  the  fork  ? 


CHAPTER  VIII. 
LIGHT. 

140.  The  Sensation  of  Light.  —  Light  is  a  sensation 
which   is  peculiar  to  the  optic  nerve.     It  may  be  pro- 
duced in  various  ways,  but  is  usually  the  result  of  dis- 
turbances in  external  objects  which  are  transmitted  to 
the  eye  as  waves.     What  we  have  learned  of  Avaves  in 
general  and  of  sound  waves  will  be  of  use  to  us  in  the 
study  of  light.     It  is  common  to  call  by  the  name  light 
the  cause  of  the  sensation  of  light  as  well  as  the  sensa- 
tion itself,  and  no  serious  confusion  need  result  from 
our  using  the  term  in  both  senses. 

141.  Definitions.  —  Every  body  which  is  at  any  in- 
stant visible  to  us  is  said  to  emit  light  at  that  instant. 
A   small  class  of  bodies  emit   light   regardless    of  the 
presence  or  absence   of  other  luminous  bodies.     Such., 
bodies  are  said  to  be  self-luminous.     They  are  usually, 
though  not  always,  very   hot,    like  the    sun  or  a  gas 
flame.     Bodies  through  which  objects  are  clearly  visible 
are  transparent,  as  water  and  glass.     Bodies  which  are 
not  transparent  but  yet  alloAV  much  light  to  pass  through 
are    translucent.     Such   are    milk,   ground   glass,    opal. 
Bodies  which  allow  no  light  to  pass  are  opaque.     Most 
solid  bodies  (not  crystals)  are  of  this  class. 


LIGHT  FROM  A   POINT.     SHADOWS. 


227 


142.  Light  from  a  Point.  Shadows. — Every  lu- 
minous body  may  be  thought  of  as  made  up  of  a  large 
number  of  luminous  points.  If  light  is  a  disturbance 
which  spreads  from  a 
luminous  point  in  a  uni- 
form medium  it  will 
advance  with  equal  ve- 
locities in  all  directions 
until  it  meets  a  differ- 
ent medium.  Let  tS 
(Fig.  165)  be  such  a  FIG.  165. 

luminous  point.  The  wave  fronts  are  concentric  spheres 
and  are  always  perpendicular  to  radii  drawn  from  S  as  a 
centre.  An  opaque  object  as  0  0'  cuts  off  all  that  portion 
of  the  waves  which  strikes  it,  leaving  a  space  POOP1 

which  is  not  illumi- 
nated by  /SI  This 
agrees  with  what  we 
have  all  observed. 
It  is  commonly  ex- 
pressed by  saying 
that  light  travels  in 
straight  lines.  The 
normal  to  a  wave  front  is  often  called  a  "ray."  We 
shall  hereafter  understand  the  word  ray  in  that  sense. 
The  shadow  cast  by  a  luminous  surface  has  not  sharp 
outlines,  since  there  are  portions  which  lie  in  the  shad- 
ows of  some  points  and  yet  are  illuminated  by  other 
points.  Fig.  166  shows  the  shadow  cast  by  a  luminous 


228 


LIGHT. 


surface.  There  are  supposed  to  be  an  unlimited  num- 
ber of  points  between  S  and  /S\.  The  space  between 
P  and  Q  receives  light  from  an  increasing  number  of 
these  points  as  we  proceed  from  P  to  Q.  The  space 
between  P  and  Q'  receives  no  light  from  >S>Sr  It  is 
called  the  umbra,  or  full  shadow.  The  rest  of  the  space 
between  Q  and  P'  is  called  the  penumbra,  or  partial 
shadow.  During  a  partial  eclipse  of  the  sun  that  part 
of  the  earth  where  the  eclipse  is  seen  passes  into  the 
penumbra  of  the  shadow  cast  by  the  moon.  During  a 
total  eclipse  the  observer  is  in  the  umbra. 

143.  Pictures   by   Small  Openings.  —  A    beautiful 
illustration    of   the  rectilinear  propagation    of  light   is 

furnished  by 
making  a  single 
small  opening 
thro  u  g  h  the 
side  of  a  box 
FIG.  167.  '  and  fastening 

a  sheet  of  paper  against  the  side  opposite  the  opening. 
If  we  follow  the  light  from  two  points  of  an  object,  as 
SSl  (Fig.  167),  till  it  strikes  the  paper,  we  see  that  for 
every  point  of  the  object  there  is  a  corresponding  spot 
on  the  paper.  The  result  is  to  give  us,  on  the  paper,  an 
inverted  picture,  or  image  as  it  is-  called,  of  the  object. 
The  image  is,  however,  faint  if  the  hole  is  very  small, 
and  blurred  if  the  hole  is  larger,  for  the  spots  S'SJ  are 
not  points,  but  circles  larger  than  the  opening. 


PICTUEES  BY  REFLECTION. 


229 


All  our  perception  of  the  forms  of  objects  is  condi* 
tioned  upon  the  formation  of  images  upon  the  sensitive 
wall  (retina)  of  the  eye.  We  shall  now  explain  the 
ways  in  which  images  may  be  formed  which  are  much 
more  perfect  in  detail  as  well  as  more  brilliantly  illumi- 
nated than  the  pinhole  images  just  described. 

144.  Pictures  by  Reflection.  —  We  learned  in  our 
study  of  waves  that  a  wave  after  reflection  from  a 


FIG.  168. 

plane  surface  makes  an  angle  equal  to  the  angle  of 
incidence,  but  in  the  opposite  direction.  Every  reflect- 
ing surface  is  plane  for  a  very  small  area  at  any  point  on 
the  surface.  The  angle  made  by  a  wave  at  any  point 
on  a  curved  surface  with  the  surface  at  that  point  is  the 
angle  which  it  makes  with  a  plane  tangent  to  the  surface 
at  the  point.  Let  us  bear  in  mind  that  the  difficulty  with 
our  pinhole  image  is  that  the  light  from  AS'  which  falls 
exactly  at  8'  (Fig.  167)  is  only  that  which  travels  along 


230 


LIGHT. 


the  radius  SS'9  and  if  we  attempt  to  get  more  light  by 
making  the  hole  larger,  none  of  the  additional  light 
admitted  falls  exactly  upon  S',  but  only  near  it,  making 
a  confused  image.  The  problem  is,  then,  to  make  a 
large  portion  of  the  waves  winch  diverge  from  S  meet 
at  S'.  This  can  only  be  done  by  changing  the  direction 
of  the  waves  in  some  regular  manner,  such  that  those 
which  diverge  most  from  the  radius  SS1  shall  be  bent 


M' 


more  than  those  which  diverge  less.  This  is  accom- 
plished if  the  waves  are  reflected  from  a  spherical  sur- 
face MM',  as  in  Fig.  168,  where  all  the  waves  from  S 
which  strike  the  mirror  meet  at  $',  which  is  called  the 
image  of  8.  The  light  after  reflection  seems  to  an  eye 
at  E  to  diverge  from  S'.  A  series  of  such  images 
arranged  symmetrically  with  corresponding  points  in 
the  object  'bofistitute  a  real  image  of  the  object.  When 
the  mirror  is  of  such  a  form  that  the  waves  do  not 


PICTUEE8  BY  REFLECTION. 


231 


really  diverge  from  centres,  but  only  seeni  to  do  so,  the 
image  is  said  to  be  virtual.  '  Thus  the  convex  mirror  in 
Fig.  169  makes  the  waves  from  8  diverge  still  more 
than  they  did  before  striking  the  mirror,  but  they  seem 
to  diverge  from  a  point  S'  behind  the  mirror  where 
there  are  really  no  waves  from  S,  and  3'  is  said  to  be  a 
virtual  image  of  S. 

A  plane  mirror  also  gives  a  virtual  image,  but  the 


amount  of  divergence  of  the  waves  is  unchanged 
(see  Fig.  170).  It  is  easily  shown  that  the  image  of  a 
point  in  a  plane  mirror  lies  in  a  line  drawn  from  the 
point  perpendicular  to  the  mirror  and  as  far  beliind  the 
mirror  as  the  point  is  in  front.  Let  $'  {  T7ig.  171)  be 
the  image  of  S  in  a  plane  mirror,  and  $J^a  ^perpendic- 
ular to  the  mirror.  S'  must  lie  on  the  line  SP  pro- 


232 


LIGHT. 


duced,  since  SP  is  at  once  perpendicular  to  the  mirror 
and  the  wave  front,  making  the  angle  of  incidence  zero. 
Let  SQ  be  the  direction  of  another  portion  of  the  wave 
which  strikes  the  mirror  at  Q  and  draw  RQ  perpendicu- 


FIG.  171. 

lar  to  the  mirror  at  Q.     Let  QSn  be  the  direction  of 
propagation  of  the  wave  after  reflection  from  Q. 
Angle  PSQ    =  Angle  SQR 
„      SQR    =      „     RQS" 
„     EQS»=      „     PS'Q 
whence:     „      PSQ    =      „     PS'Q 

In  the  right  triangles  SPQ  and  S'PQ  the  sides  PQ  are 
identical,  and  angles  PSQ  and  PS'Q  are  $qual.  The 
triangles,  therefore,  are  equal  and  PS1  =  PS. 

145.  Position  of  the  Image  of  an  Object.  —  To  de- 
termine the  position  of  the  image  of  an  object  we  may 


POSITION  OF  THE  IMAGE  OF  AN  OBJECT.      233 


locate  the  image  of  two  or  more  prominent  points.  In 
the  case  of  plane  mirrors  this  is  easily  done  by  drawing 
perpendiculars  through  the  mirror  to  points  as  far  be- 
hind the  mirror  as  the  luminous  points  are  in  front 
of  it.  In  the  case  of  spherical  mirrors  it  might  be 
done  by  drawing  any  two  radii  till  they  meet  after 
reflection.  The  measurement  of  angles  is,  however,  less 
easy  than  the  measurement  of  lines,  and  it  happens  that 
there  are  two  radii  (or  rays  as  they  are  commonly  called) 
which  can  always  be  drawn  without  measuring  the 
angles.  Let  MM' 
(Figs.  172  and 
173)  be  a  spherical 
mirror  drawn  from 
C  as  a  centre.  A 
line  through  0 
perpendicular  t  o 
the  centre  of  the 
mirror  is  called  the  FIG- 172- 

principal  axis  of  the  mirror.  All  waves  passing  along 
a  ray  drawn  through  C  are  reflected  back  along  the  same 
ray.  It  can  be  proved  that  all  waves  moving  along 
rays  drawn  parallel  to  the  principal  axis,  CX,  will  be 
reflected  along  rays  which  pass  through  a  point  jP,  half- 
way between  O  and  the  mirror.  This  point  is  called 
the  principal  focus.*  It  is  the  place  where  plane  waves, 
like  those  from  the  sun  or  any  distant  object,  meet  after 
reflection.  Let  it  be  required  to  find  the  image  of  an 

*  Latin  focus,  fireplace. 


234 


LIGHT. 


object  SSr  Draw  SPF,  making  SP  parallel  to  OX. 
Draw  SCQ.  Their  point  of  intersection  S'  is  the  image 
of  8.  Find  the  image  of  ^  in  the  same  manner.  Join 
them  and  we  have  the  image  of  the  object.  For  differ- 
ent positions  of  the  object  the  position  of  the  image 
will  be  different,  but  the  method  of  finding  the  position 
of  the  image  is  always  the  same. 

The  statement  that  all  rays  parallel  to  the  principal 
axis  pass  through  the  principal  focus  is  only  true  if  the 
mirror  is  a  small  segment  of  a  sphere.  Large  mirrors 


FIG.  173. 


must  have  a  correspondingly  large  radius  of  curvature. 
Spherical  mirrors  are  but  little  used  in  optical  instru- 
ments, the  reflecting  telescope  being  almost  the  only 
example  of  their  use.  They  are  much  employed  for 
collecting  and  directing  the  light  of  signal  lamps,  loco- 
motive headlights,  and  the.  like,  where  no  accurate 
image  is  required.  If  the  lamp  is  placed  at  the  focus 
of  a  concave  mirror  the  reflected  rays  will  be  parallel. 
By  moving  it  toward  the  mirror  any  desired  amount  01 
divergence  may  be  produced.  By  moving  it  away  from 


PICTURES  BY  REFRACTION. 


235 


the  mirror  the  light  may  be  made  to  converge  at  any 
desired  point. 

146.  Pictures  by  Refraction.  —  We  have  seen  in 
Section  127  that,  in  general,  waves  suffer  a  change  of 
direction  in  passing  from  one  medium  to  another.  . 
Let  LL  (Fig.  174)  be  a  segment  of  a  sphere,  of 
glass  say,  made  by  a  plane  passing  at  some  distance 
from  the  centre,  so  that  it  shall  be  but  a  small  seg- 
ment of  a  sphere,  or  by  the  intersection  of  two 
spheres.  It  is  a  lens.  When  convex  on  both  sides, 
it  is  called  a  double-convex  lens.  Light  waves  from 
S  which  fall  upon  LL  (Fig.  175)  will  be  in  part  re-  -L 
fleeted,  but  we  are  at  present  concerned  with  the  174.' 
part  that  passes  through  the  lens.  The  waves  strike  the 
central  part  of  the  lens  first,  and  are  retarded,  since 


the  velocity  in  glass  is  less  than  in  air.     In  the  case 

shown  in  Fig.  175  the  waves  in  the  glass  are  nearly 

'  plane.     The  portions  of  the  wave  farthest  from  the  axis, 

(7(7,  emerge  first  and  gain  upon  the  part  that  has  not 


236 


LIGHT. 


yet  emerged,  so  that  when  the  wave  is  again  in  air  its 
curvature  has  been  reversed,  and  it  converges  at  S'. 

All  lenses  which  are  thicker  at  the  centre  than  at  the 
edge   converge  the  waves    and    are    called    converging 

lenses.  Three  kinds  of  con- 
verging lenses  are  shown  in 
Fig.  176,  namely:  (1)  double- 
convex,  (2)  plano-convex, 
(3)  concavo-convex. 

The  principal  focus  of  a 
converging  lens  is  easily  found 
by  experiment.  It  is  the  place 
of  convergence  of  plane 
waves  which  have  passed  through  the  lens.  In  a  double- 
convex  lens  of  flint  glass  with  equal  curvatures  on  its 


FIG.  177. 

two  sides,  the  focus  is  not  far  from  the  centre  of  cur- 
vature. The  distance  OF  (Fig.  177)  is  called  the  focal 
length  of  the  lens. 

An  object  placed  at  a  distance  2  OF  from  the  lens 
will  be  focused  at  an  equal  distance   on  the  opposite 


IMAGES  FORMED  BY  LENSES. 


237 


FIG.  178. 


side  at  F',  which  is  sometimes  called  the  secondary 
focus. 

Objects  outside  F'  will  be  focused  between  F  and  F', 
and  objects  between  F  and  the  lens  will  be  focused  out- 
side F'. 

Lenses 
which  are 
thicker  at  the 
edges  than  in 
the  centre  in- 
crease the  di- 
vergence of 

waves  and  are  called  diverging  lenses.  Since  they  can- 
not cause  divergent  waves  to  converge,  they  never  form 
real  images,  but  only  virtual  ones  (see  Fig.  178). 

Three  forms  of  diverging  lenses  are  shown  in  Fig.  179, 

namely:  (1)  convexo- 
concave,  (2)  plano-con- 
cave, (3)  double-concave. 
Diverging  lenses  are 
used  by  draughtsmen  to 
enable  them  to  judge 
how  a  drawing  will  ap- 
pear when  reduced  in 
size.  They  are  used  also 
in  spectacles  for  near-sighted  persons.  Most  lenses  used 
in  optical  instruments  are  converging. 

147.  Position  of  Images  Formed  by  Lenses.  —  As 
in  mirrors,  so  in  lenses,  if  we  find  where  two  rays  from 


238 


LIGHT. 


a  point  meet  again  we  have  found  the  image  of  that 

point. 

Rays  parallel  to  the 
principal  axis  are  re- 
fracted so  as  to  converge 
at  the  focus.  Waves 
which  pass  through  0, 
the  optical  centre  of  the 
lens,  suffer  no  perma- 
nent change  of  direction, 
but  only  a  slight  dis- 
placement to  one  side, 
like  light  which  has 

FJG  18()  passed   through  a  plate 

with  parallel  sides  (see 


FIG.  181. 


Fig.  180),  for  the 
faces  -at  points 
intersected  by  a 
line  through  the 
centre  are  par- 
allel. - 

To  find  the 


FIG.  182. 


SOME   OPTICAL  INSTRUMENTS. 


239 


image  of  an  object,  SS^  (Figs.  181  and  182),  we  first 
find  the  image  of  S  by  drawing  rays  8PF  and  SQ  0  and 
producing  them  till  they  meet  at  S'.  The  image  of  S1 
is  found  in  a  similar  manner.  The  image  is  found  to  be 
real  and  inverted  in  one  case,  virtual  and  upright  in  the 
other. 

The  two  cases  shown  are  typical  and  all  other  cases 
may  be  treated  in  the  same  manner. 

148.  Some    Optical    Instruments.  —  The    simplest 
optical  instrument  is   the   magnifying  glass,  or   simple 

microscope.       1 1 

consists  of  a  con- 

verging  lens. 

The   object   is 

placed    between 

F  and  the  lens, 

and  the  image  is  FlG- 183- 

virtual,  upright  and  enlarged,  as  shown  in 

Fig.  183. 

The  compound  microscope  (Fig.  184)  con- 
sists of  a  small 

converging  lens 

of  short  focus, 
FIG.  184.  called  the  ob- 
jective, which  is  placed  near 
the  object  and  forms  an  in- 
verted real  image  at  the 
upper  end  of  the  tube,  and  FlGt 


240 


LIGHT. 


FIG.  186. 


a  second  converging  lens  called  the  eyepiece,  which  is  a 

single  or  double  magnifying  lens  used  to  view  the  image 

formed  by  the  objective. 

The  camera  consists  of  a  dark  box  having  a  lens,  L 
(see  Fig.  185),  at  the  front  which  forms  a 
real  inverted  image  on  a  screen  at  the  back. 
The  distance  between  lens  and  screen,  ss, 
may  be  varied  to  bring  objects  at  different 
distances  into  focus.  The  screen  may  be 
removed  also,  and  replaced  by  a  sensitive 
plate.  A  diaphragm,  d,  often  of  the  form 

shown  in  Fig.  186,  may  be  turned  so  as  to  control  the 

amount  of  light  admitted  through  the  lens. 

The  eye  is  very  like  a  camera.     It  has  a  lens  called 

the  crystalline   lens, 

L  (Fig.  187),  a  sen- 
sitive membrane,  the 

retina,  r,  at  the  back 

to  receive  the  image, 

a  diaphragm,  the  iris, 

i,  for  varying  the 

amount  of  light,  and 

an    arrangement  for 

adjusting    the    focus 

for  near  or  distant 

objects. 

This  adjustment  is 

called  accommodation,  and  is  accomplished  by  changing 

the  curvature  of  the  lens  by  means  of  certain  muscles 

at  its  circumference. 


FIG.  187. 


SOME    OPTICAL   INSTRUMENTS.  241 

Many  eyes  are  faulty  in  having  the  retina  too  near  to 
or  too  far  from  the  lens.  When  the  retina  is  too  far 
back  the  object  must  be  brought  near  in  order  to  be  dis- 
tinctly seen.  The  person  is  near-sighted  and  should 
wear  concave  glasses.  When  it  is  too  near  the  person 
is  far-sighted  and  should  wear  convex  glasses.  • 

When  the  normal  eye  is  at  rest  it  is  in  focus  for  dis- 
tant objects.  Near  objects  are  seen  only  by  bringing 
into  use  the  muscles  of  accommodation.  At  about  forty 
years  of  age  we  begin  to  lose  the  power  of  accommoda- 
tion and  must  wear  glasses  for  reading,  though  distant 
objects  are  still  seen  distinctly. 

There  are  other  errors  of  vision  which  the  skilled 
oculist  can  correct.  One  who  has  any  difficulty  with 


c  P 


FIG.  188. 


his  eyesight  should  consult  an  oculist  without  delay. 
Errors  of  vision  cause  headache  and  may,  if  uncorrected, 
lead  to  inflammation  of  the  eye  and  serious  impairment 
of  vision. 

The  magic  lantern,  or  stereopticon,  consists  of  a  pair 
of  lenses  called  the  condenser,  0  (Fig.  188),  the  pur- 
pose of  which  is  to  collect  the  light  from  a  bright 


242  LIGHT. 

source,  S,  like  an  arc  lamp,  and  make  it  converge  upon 
a  projection  lens,  L.  Between  C  and  L  and  very  close 
to  the  former  is  placed,  inverted,  the  transparent  pic- 
ture, jt?,  an  image  of  which  is  thus  projected  upon  a 
large  screen  at  ss. 

COLOR. 

Up  to  this  point  we  have  considered  only  the  forms 
of  images.  We  all  know,  however,  that  the  color  of  an 
object  has  often  as  much  to  do  with  its  appearance  as 
the  form.  The  image  formed  in  the  camera  is  a  col- 
ored image,  though  the  ordinary  photograph  reproduces 
only  the  lights  and  shades  of  the  picture  and  not  its 
color. 

The  eye,  unlike  the  photographic  plate,  distinguishes 
differences  in  the  appearance  of  objects  which  have 
exactly  the  same  shape,  as  when  ripening  fruit  or 
autumn  leaves  change  from  green  to  red. 

According  to  Helmholtz,  the  eye  is  sensitive  to  three 
fundamental  impressions,  namely,  red,  green,  and  violet. 
The  hundreds  of  colors  which  we  are  able  to  distinguish 
result  from  the  blending  of  these  three  fundamental 
sensations  in  varying  proportions.  What  we  call  white 
light  is  the  mixture  of  equal  parts  of  the  three. 

149.  Color  by  Refraction.     The  Spectrum.  —  The 

compound  nature  of  white  light  was  shown  by  Newton 
in  a  beautiful  experiment  which  may  easily  be  repeated 
by  any  one.  Sunlight  is  admitted  to  a  darkened  room 


COLOR  BY  REFRACTION.     THE  SPECTRUM.     243 

through  a  small  hole  in  a  shutter  and  allowed  to  fall 
upon  a  prism  of  glass,  as  shown  in  Fig.  189.  The  light 
will  be  refracted,  as  we  know  already,  but  it  will  not 
all  be  refracted  to  the  same  extent.  The  result  is  to 
give  us  a  row  of  images  of  the  opening,  forming  a  beau- 
tiful band  of  colors  called  a  spectrum. 

If  the  opening  is  circular  the  images  will  overlap 
and  the  colors  will  be  mixed,  but  if  a  narrow  slit  is  used 
the  colors  are  pure. 

The   rainboiv  is  a  great   spectrum  produced    by  the 


FIG.  189. 

refraction  of  light  in  drops  of  water.  Close  observation 
Avill  show  that  the  arrangement  of  the  colors  is  the  same 
as  in  Newton's  spectrum. 

It  is  now  known  that  the  difference  in  light  waves 
which  causes  a  difference  in  refrangibility  and  also  in 
color  is  one  of  wave  length.  The  unit  of  wave  length 
is  the  millionth  of  a  millimetre.  The  shortest  visible 
waves  are  the  violet,  380  units  long,  and  the  longest 
are  the  red,  reaching  to  688.  Between  these  limits  are 
waves  of  every  possible  length. 


244 


LIGHT. 


The  violet  sensation  is  excited  most  by  waves  in  the 
neighborhood  of  450,  but  as  far  as  380  on  the  one  side 
and  560  on  the  other.  The  relative  effect  of  the  differ- 
ent wave  lengths  in  producing  the  violet  sensation  is 
shown  by  the  height  of  curve  F(Fig.  190). 

The  green  sensation  is  excited  most  strongly  by  550, 
but  responds  to  Avaves  from  440  to  640  (see  6r). 

The  maximum  effect  of  the  red  is  at  570,  but  it  ex- 
tends as  far  as  480  and  688  (see  7T). 

It  will  be  seen  that  the  waves  between  480  and  560 

excite  to  some 
extent  all 
three  of  the 
primary  s  e  n- 
sations  in  the 
retina  of  the 
eye.  The  ef- 
fect of  excit- 
ing the  violet 
and  the  green 

at  the  same  time  is  to  give  us  the  sensation  of  blue, 
which  may  be  a  deep  indigo  if  but  little  green  is  present, 
or  an  emerald  if  but  little  violet  is  present. 

In  the  same  way  yellow  and  orange  result  from  mix- 
ing the  sensations  of  green  and  red. 

The  mixture  of  red  and  violet  gives  purple,  a  color 

which  does  not  appear  in  the  spectrum,  since  there  are 

no  waves  which  excite  both  red  and  violet,  and  not  green. 

Any  pure  hue  mixed  with  white  gives  tints;  mixed 


700 


650 


550 


500 


150 


350 


Draw    Trtlot        Green        Blu»     Indigo     Vlolii 


FIG.  190. 


COLOR  MIXTURE.  245 

with   black,   shades.     Thus   the  browns   are    shades   of 
orange  and  sky  blue  is  a  tint  of  blue. 

150.  Color  Mixture.  —  If  a  circular  disc  be  divided 
into  three  equal  sectors,  each  of  which  is  painted  one  of 
the  primary  colors,  red,  green,  violet,  as  in  Fig.  191,  the 
disc  will  show,  when  rapidly  rotated  about  its  centre,  no 
one  of  these  colors,  but  will  appear  gray.  If  the  colors 
were  perfectly  pure  it  would  appear  white,  but  the  black 
in  them  gives  the  mixture  a  gray  appearance.  A  smaller 
disc  having  white  and  black  segments  may  be  placed 
over  the  colored  disc,  and,  if  the  black  and  white  are  in 
the  right  proportions,  the  two  grays  will  match. 

The  reason  that  the  separate  colors  are  not  seen  when 
the  disc  is  in  motion  is  that  any  impression  produced 
upon  the  retina  persists  for  a  fraction  of  a  second,  so 
that  each  of  the  colors  in  the  rotating  disc  affects  the 
retina  the  same  as  if  the  others  were  not  present.  The 
result  is  that  three  colored  images  fall  upon  the  same 
spot  of  the  retina,  and,  since  these  colors  excite  all  the 
sensations  that  white  light  is  able  to  excite,  and  in  the 
same  proportions,  the  effect  produced  is  that  of  white 
light.  If  the  colors  were  as  bright  as  those  of  the  spec- 
trum the  mixture  would  be  white.  Since  the  colors  are 
all  more  or  less  mixed  with  black,  it  is  in  reality  gray. 

A  convenient  plan  for  mixing  the  colors  in  any  de- 
sired proportions  is  to  make  a  number  of  discs  like  the 
one  shown  in  Fig.  192,  and  paint  each  a  different  color. 
The  discs  have  each  a  radial  slit  so  that  they  may  be 


246  LIGHT. 

slipped  one  upon  the  other,  as  shown  in  Fig.  193.  They 
may  then  be  turned  so  as  to  expose  to  view  any  desired 
proportions  of  the  different  colors. 

Another  method  of  mixing  the  sensations  is  to  rule 
alternate  lines  of  the  colors  very  close  together,  as  in 
Fig.  194.  The  images  are  not  kept  distinct  by  the  eye, 
but  blend  together  and  give  the  composite  color,  orange 
or  blue. 

Experiment  shows  that  a  mixture  of  red  and  green- 
ish blue  produces  gray.  This  should  be  so,  since  blue 
is  composed  of  green  and  violet.  Similarly,  yellow  and 
indigo  blue  produce  gray,  since  together  they  contain  all 
the  primary  colors. 

Such  pairs  of  colors  are  called  complementary  colors. 
A  number  of  such  pairs  are  shown  in  the  following 
table :  - 

COLOR  COMPLEMENTARY 

Red  Greenish  blue 

Orange  Blue 

Yellow  Indigo 

Green  Purple 

Bluish  green  Red 

Blue  Orange 

Indigo  Yellow 

Violet  Greenish  yellow 

The  colors  made  by  mixing  pairs  of  spectrum  colors 
are  shown  in  the  following  table,  where  the  mixture  of 
any  color  named  in  the  top  row  with  any  color  in  the 
left-hand  column  will  be  found  at  the  intersection  of 
the  corresponding  row  and  column: 


FIG.  191. 


FIG.  192. 


FIG.  193. 


FIG.  194. 


COLORS  BY  ABSORPTION. 


247 


Violet 

Indigo 

Blue 

Blue 
Green 

Green 

Yellow 

Red 

Purple 

Dark 
Rose 

Light 
Rose 

White 

Light 
Yellow 

Orange 

Orange 

Dark 
Rose 

Light 
Rose 

White 

Light 
Yellow 

Yellow 

Yellow 

Green 
Yellow 

Light 
Rose 

White 

Light 
Green 

Light 
Green 

Green 
Yellow 

White 

Light 
Green 

Light 
Green 

Green 

Green 

Light 
Blue 

Green 
Blue 

Blue 
Green 

Blue 
Green 

Green 
Blue 

Green 
Blue 



Blue 

Indigo 

151.  Colors  by  Absorption.  Pigments.  — What  we 
call  the  color  of  an  object  is  the  color  which  it  shows 
when  illuminated  by  white  light.  The  reason  that  dif- 
ferent objects  have  different  colors  is  that  they  do  not 
reflect  all  colors  in  the  same  proportion. 

When  white  light  falls  upon  a  red  brick  wall  the 
waves  which  produce  green  and  violet  sensations  are 
absorbed  by  the  brick  and  converted  into  heat,  while 
the  waves  which  produce  the  red  sensation  reach  the 
eye.  In  like  manner  an  orange  absorbs  the  violet  and 
reflects  green  and  red  in  about  equal  proportions,  while 
a  lemon  absorbs  'Some  of  the  red  as  well  as  the  violet, 
giving  us  a  greenish  yellow. 

A  black  object  is  one  which  absorbs  all  the  waves, 


248  LIGHT. 

while  a  gray  object  is  one  which  absorbs  about  the  same 
part  of  each  of  the  primary  colors. 

We  may  now  understand  why  the  colored  discs  pro- 
duce gray  and  not  white.  It  is  because  the  red  disc 
absorbs  not  only  the  green  and  violet,  but  a  considerable 
portion  of  the  red  as  well.  The  same  being  true  of  the 
other  colors,  the  mixture  must  contain  a  large  amount 
of  black. 

When  two  paints  are  mixed  the  result  is  very  differ- 
ent from  that  obtained  by  mixing  the  sensations  pro- 
duced by  the  separate  paints.  The  reason  for  this  is 
that  the  color  which  would  be  transmitted  by  one  pig- 
ment is  absorbed  in  part  by  the  other,  and  only  that  is 
transmitted  which  neither  pigment  absorbs. 

A  yellow  disc  and  a  blue  one  will  give  gray  when 
blended  by  rotation,  but  yellow  paint  and  blue  paint 
mixed  give  green,  for  the  blue  absorbs  red  and  trans- 
mits violet  and  green,  while  the  yellow  absorbs  violet 
and  transmits  red  and  green.  The  two,  therefore,  ab- 
sorb all  but  the  green.  The  process  may  be  expressed 
by  an  equation  thus : 

(a)    Yellow  =  Bed  +  Green 
(6)        Blue  ==  Violet  -f  Green 

(a)  -f  (6)    Yellow  +  Blue  =  Bed  +  Green  +  Violet  +  Green 
=  White  -f  Green 

The  fact  that  green  paint  may  be  made  by  mixing 
blue  and  yellow  paints  led  artists  to  reckon  blue  and 
yellow  as  primary  colors,  and  green  as  a  mixed  color. 


COLORS  BY  INTERFERENCE.  249 

The  classification  suits  the  convenience  of  the  artists, 
who  continue  to  employ  it. 

Most  of  the  greens  employed  by  artists  are,  however, 
not  'made  by  mixing  blue  and  yellow,  but  are  simple 
pigments  like  the  oxide  of  chromium,  and  many  com- 
pounds of  copper. 

152.  Colors  by  Interference.  —  Light  waves,  like 
water  waves,  sound  waves,  and  all  other  waves,  may 
interfere  so  as  alternately  to  reenforce  or  destroy  each 
other. 

When  light  from  any  source  reaches  a  given  point  by 
two  paths  which  differ  in  length,  it  will  in  general  in- 
terfere. If  the  light  is  of  a  single  color,  as  red,  the 
waves  which  have  come  by  the  two  paths  will  destroy 
each  other  if  the  difference  in  length  of  the  paths  is  a 
half  wave  length  or  any  whole  number  of  half  wave 
lengths  of  red  light.  They  will  reenfore  each  other  if 
the  difference  of  path  is  a  whole  wave  length  or  any 
whole  number  of  wave  lengths. 

Suppose  plane  waves  of  wave  length,  I,  to  enter  a 
dark  box  by  two  small  openings,  0,  0'  (Fig.  195),  which 
are  very  close  together,  and  fall  upon  a  screen  at  N. 
The  openings  0,  0'  will  act  like  sources  of  light,  send- 
ing waves  in  every  direction  in  the  box. 

If  P  be  a  point  on  the  screen  at  such  a  distance  from 
0  that  O'P—  OP=  J  Z,  there  will  be  a  dark  band  at 
P  where  the  waves  meet  in  opposite  phase  and  destroy 
each  other. 


250 


LIGHT. 


If  Q  be  a  point  such  that  O'Q  —  OQ  =  7,  the  waves 
will  reenforce  each  other,  and  a  bright  band  will  be 
seen. 

Had  the  light  been  white  instead  of  being  light  of  a 
single  color,  Q  would  have  had  a  different  position  for 
each  different  wave  length,  and  the  band  would  have 
been  spread  out  into  a  spectrum.  The  violet  end  would 


FIG.  195. 

lie  nearest  the  point  N,  the  point  where  the  normal  to 
the  wave  before  it  entered  the  box  meets  the  screen. 
This  bending  of  a  wave  when  passing  through  small 
openings  is  called  diffraction.  A  spectrum  produced  by 
diffraction  is  called  a  diffraction  spectrum.  In  the  dif- 
fraction spectrum  the  dispersion  is  proportional  to  the 
wave  length,  and  the  colors  are  arranged  as  indicated 
in  Fig.  190,  where  the  green  is  seen  to  occupy  the 


THE  SPECTEOSCOPE.  251 

middle  of  the  spectrum.  In  the  refraction  spectrum 
the  violet  end  is  dispersed  much  more  than  the  red 
end  (see  Fig.  198). 

A  candle  viewed  through  a  slit  in  a  card  will  show 
diffraction  spectra.  The  candle  should  be  several  me- 
tres distant  in  a  dark  room  and  the  card  should  be  held 
near  the  eye.  The  slit  may  be  made  with  the  point  of 
a  sharp  knife  in  the  middle  of  a  visiting  card. 

An  umbrella  held  toward  an  arc  lamp  will  show  dif- 
fraction colors.     The  moon  viewed  through  a  window    % 
screen  appears  drawn  out  in  the  form  of  a  cross. 

153.  Colors  of  Thin  Plates. — When  white  light  is 
reflected  from  two  plane  surfaces  which  are  very  near 
together  the  waves  from  one  of  the  surfaces  may  inter- 
fere with  those  from  the  other  surface,  so  as  to  destroy 
light  of  a  certain  wave  length.     The  reflected  light  will 
show   the    color   which   is    complementary  to    the    one 
destroyed. 

The  colors  seen  on  a  soap  bubble  are  produced  in  this 
way,  as  are  also  the  gay  colors  of  many  insects  and  birds. 
The  feathers  of  a  peacock's  tail  would  lose  their  brilliant 
colors  if  they  were  pressed  flat  enough  to  destroy  the 
uniform  arrangement  of  the  little  barbs  of  which  they 
are  composed.  Mother-of-pearl  ground  to  powder  loses 
all  its  bright  tints  and  appears  like  chalk  dust. 

154.  The   Spectroscope.  —  For  a  careful   study   of 
spectra  an  instrument  called  a  spectroscope  is  used.     It 


252 


LIGHT. 


consists  of  a  stand  carrying  a  circular  plate,  on  which  is 
mounted  a  tube  called  the  collimator,  C  (Figs.  196  and 
197),  which  has  an  adjustable  slit  at  the  end  nearest  the 
source  of  light,  and  a  lens  at  the  other  end  to  make 
parallel  the  rays  of  light  before  they  fall  upon  the 
prism,  P  (Fig.  196),  or  the  grating,  G-  (Fig.  197),  which 


FIG.  196. 


FIG.  197. 


is  to  produce  the  spectrum.  The  spectrum  is  viewed 
by  the  telescope,  T,  which  is  mounted  to  point  toward 
the  centre  of  the  circle. 

Outside  light  may  be  cut  off  by  a  black  cloth  laid 
over  the  central  part  of  the  instrument.  The  instru- 
ment is  usually  graduated  to  degrees,  so  that  the 
amount  of  deviation  may  be  measured. 


SPECTEUM  ANALYSIS.  253 

155.  Spectrum  Analysis.  —  The  spectra  shown  by 
different  bodies  differ  very  much.  They  may,  however, 
be  divided  into  three  classes : 

I.  Band  Spectrum.  —  This  is  the    spectrum   shown 
by  any  white-hot  solid  or  liquid.     A  gas  flame  shows 
such  a  spectrum,  since  the  luminous  part  of  the  flame  is 
composed  of  solid  carbon  particles.     It  is  these  parti- 
cles of  solid  carbon  that  collect,  as  lampblack,  upon  any 
cold  object  held  for  an  instant  in  the  flame.     The  band 
spectrum  shows  all  the  colors  of  the  rainbow,  that  is  to 
say,  it  is  produced  by  light  of  all  wave  lengths. 

II.  Bright   Line  Spectrum.  —  This  is  the  spectrum 
of  a  glowing  gas,  like  that  obtained  by  burning  salt  in 
the  flame  of  an  alcohol  lamp    or    Bunsen    burner.     It 
consists  of  one  or  more  narrow  lines  which  are  always 
in  the  same  relative  position  for  a  given  substance,  be- 
cause in  a  gas  the  particles  are  free  to  vibrate  as  they 
please,  and  the  atoms  of  each  substance  seem  to  have  a 
definite  period  of  vibration  which  gives  rise  to  light  of 
corresponding  wave  length,  just  as  a  given  rod  or  string 
gives  rise  to  sound  waves  of  a  particular  pitch. 

In  solids  the  particles  jostle  against  one  another  and 
so  vibrate  in  all  possible  periods  (Class  I). 

The  bright  line '  spectrum  of  a  substance  gives  the 
chemist  a  means  of  detecting  the  presence  in  a  com- 
pound of  very  minute  portions  of  a  substance.  The 
process  is  called  spectrum  analysis. 

III.  Dark  Line  or  Band  Spectrum.  —  When  white 
light  is  passed  through  a  colored  glass  or  solution  or 


254 


LIGHT. 


vapor,  some  of  the  vibrations  are  absorbed,  leaving  the 
transmitted  light  wanting  in  certain  wave  lengths.  The 
spectrum  will  appear  crossed  with  black  lines  or  bands. 
Y  Sunlight  and  the  light  from  many  stars  give  spectra  of 
this  class. 

156.  The  Solar  Spectrum.  —  Fraunhofer  was  the 
first  to  explain  the  dark  lines  of  the  solar  spectrum, 
which  are  now  known  as  Fraunhofer  s  lines.  He 
showed  that  a  vapor  will  absorb  waves  of  the  same 
length  as  those  which  it  emits.  Thus  sodium  vapor 
placed  between  an  arc  light  and  a  spectroscope  will 


A  a   B  C 


Eb 


Red       Orange  Yellow  Green 


Blue  Indigo 

FIG.  198. 


produce  a  dark  line  in  the  yellow  in  exactly  the  posi- 
tion where  the  yellow  bright  line  appears  when  we 
view  burning  sodium. 

The  dark  lines  of  the  solar  spectrum  are,  then,  due  to 
the  presence  of  substances  like  sodium,  iron  and  many 
others,  in  vapor  form,  in  the  atmosphere  of  the  sun. 

The  principal  Fraunhofer  lines,  designated  by  Fraun- 
hofer with  the  letters  of  the  alphabet,  are  indicated  in 
Fig.  198. 

157.  Velocity  of  Light. — The  speed  with  which 
light  waves  travel  through  space  is  so  enormously  great 


INTENSITY  OF  LIGHT.  255 

that  it  is  difficult  for  us  to  form  any  conception  of  it. 
For  all  ordinary  purposes  we  treat  it  as  being  infinitely 
great.  It  was  first  measured  by  Roemer,  a  Danish  as- 
tronomer, in  1675,  by  means  of  the  difference  in  the 
apparent  times  of  revolution  of  Jupiter's  moons,  depend- 
ing upon  whether  the  earth  is  moving  toward  or  away 
from  Jupiter.  He  found  it  took  light  about  1,000  sec- 
onds to  cross  the  earth's  orbit  (186,000,000  miles), 
which  gives  for  the  velocity  of  light  186,000  miles  per 
second. 

A  light  wave  would  travel  a  distance  equal  to  two  or 
three  times  around  the  earth  in  the  time  it  takes  to  wink. 
The  light  of  the  moon  reaches  us  in  a  second  and  a  half. 

158.  Intensity  of   Light. — If   three  square  cards, 
(7,  Z>,  .#  (Fig.  199),  having  heights  of  10,  20,  and  30  cen- 
timetres  respec- 
tively, be  placed 
at   distances    of 
30,  60,   and  90 

centimetres  from  ^fV  "^" 

a  source  of  light, 

S,    it    will    be      S~~  ~~c "  D 

found    that   the  FIG.  199. 

shadow  of  0  will  exactly  cover  D  or  K  This  is  equiv- 
alent to  saying  that  the  light  which  fell  on  100  square 
centimetres  at  a  distance  of  30  is  distributed  over  400 
square^ centimetres  at  twice  the  distance  and  900  square 
centimetres  at  three  times  the  distance. 


256  LIGHT. 

Light  is  thus  seen  to  follow  the  same  law  of  inverse 
squares  that  gravitation,  magnetic  attraction,  and  elec- 
tric attraction  follow. 

The  intensity  of  light  is  reckoned  in  candle  power. 
To  compare  the  intensity  of  two  sources  of  light  it  is 
necessary  to  find  a  point  at  such  a  distance  from  both 
sources  that  the  light  received  from  both  shall  be 
equally  intense.  The  distance  of  this  point  from  each 
source  is  then  measured  and  the  intensity  computed 
by  the  law  of  inverse  squares. 

Some  methods  of  measuring  candle  power  are  ex- 
plained in  Part  II. 

159.  Light  and  Electricity .— That  the  electric 
spark,  the  electric  arc,  and  the  electric  current  are 
sources  of  light  is  familiar  to  us  all.  There  is  a  sense, 
however,  in  which  a  gas  jet,  a  candle  flame,  nay,  even 
the  sun  itself,  is  an  electric  light,  for  light  waves  are 
now  known  to  be  merely  a  particular  class  of  electric 
waves  of  suitable  wave  length  to  impress  the  nerves  of 
the  retina. 

The  sun  emits  some  waves  which  are  much  too  long 
and  others  which  are  too  short  to  affect  the  eye.  Some 
of  the  long  waves  may  be  detected  by  means  of  the 
thermometer,  some  of  the  short  ones  affect  the  photo- 
graphic plate. 

The  discovery  of  the  identity  of  light  waves  and 
electrical  waves  grew  out  of  the  mathematical  study  by 
Maxwell  of  Faraday's  researches  in  electricity. 


SPACE   TELEGRAPHY.  257 

More  than  twenty  years  after  Maxwell  published  his 
electro-magnetic  theory  of  light,  Hertz,  a  young  Ger- 
man physicist,  undertook  to  find  experimental  proof  of 
Maxwell's  theory.  After  about  eight  years  of  patient 
research  he  found  that  the  waves  sent  forth  by  the 
spark  from  an  induction  coil  set  up  sympathetic  vibra- 
tions in  a  coil  of  wire  of  suitable  dimensions  at  some 
distance  from  the  spark,  and  that  the  waves  in  his  little 
secondary  coil  would  produce  a  spark  which  gave  a 
means  of  detecting  and  measuring  the  intensity  of  the 
waves  at  various  distances  from  their  source. 

Hertz  found  that  these  long  electrical  waves  (several 
metres  in  length)  could  be  reflected,  refracted,  and 
diffracted  exactly  like  light  waves.  In  short,  he  de- 
monstrated that  Maxwell's  theory  is  correct :  that  light 
is  but  one  way  in  which  the  radiant  energy  of  a  lumi- 
nous body  may  manifest  itself. 

We  speak  of  solar  light  and  solar  heat,  because  hav- 
ing organs  of  sense  to  perceive  light  and  heat  we  are 
made  daily  conscious  of  these  particular  manifestations 
of  solar  energy.  We  should  rather  think  of  solar  radia- 
tion which  may  manifest  itself  to  our  senses  as  light 
or  heat,  which  may  produce  chemical  changes  in  the 
photographic  plate  or  in  tanning  our  skins  brown  or 
bleaching  the  color  from  fabrics,  but  which  really  con- 
sists of  electrical  waves  different  in  wave  length,  but  in 
all  other  respects  alike. 

1 60.  Space    Telegraphy.  —  Electrical    waves    have 


258  LIGHT. 

found  an  important  application  in  telegraphy  at  sea. 
Marconi  has  succeeded  in  telegraphing  to  considerable 
distances  to  and  from  vessels  without  the  aid  of  wires. 
The  messages  are  sent  with  as  little  .difficulty  during 
foggy  weather  as  at  any  other  time,  a  fact  of  great  im- 
portance to  sailors  who  are  on  a  dangerous  coast  in  bad 
weather. 

There  seems,  at  present,  little  likelihood  that  the  sys- 
tem will  replace  wires  on  land  ^  or  cables  for  long 
tances  across  the  ocean. 


161.  New  Forms  of  Radiation.  —  Lenard  and  Roent- 
gen, following  some  lines  of  research  suggested  by 
Hertz,  found  that  the  radiations  from  the  negative 
terminal  of  a  vacuum  tube  (Crookes'  tube)  have  the 
power  of  penetrating  objects  which  are  opaque  to  light 
waves  and  of  causing  a  screen  coated  with  fluorescent 
chemicals  to  glow  brightly  wherever  the  rays  strike. 
These  rays,  both  the  Lenard  rays  and  the  X-rays  of 
Roentgen,  act  upon  a  photographic  plate  when  it  is 
enclosed  in  a  light-tight  envelope. 

The  fact  that  the  bones  of  our  bodies  are  less  trans- 
parent to  the  waves  than  is  the  flesh  makes  it  possible  to 
see  a  shadow  picture  of  the  bones  of  the  hand  when 
the  hand  is  held  between  a  luminous  Crookes'  tube  and 
the  fluorescent  screen. 

If  a  photographic  plate  take  the  place  of  the  screen, 
we  obtain  a  photograph  of  the  bones  of  the  hand,  and 
of  any  foreign  body  which  may  be  imbedded  in  the 


THE  ROLE   OF  WAVE  MOTION. 


259 


flesh.     Fig.  200  is  reproduced  from  such  a  photograph, 
Bequerel  has  found  that  certain  salts  of  uranium  emit 
rays  which  have  the  same  power  of  impressing  the  plate 
that    the    X-rays 
have. 

The  subject  of 
electrical  radia- 
tions has  aroused 
great  interest 
among  physicists, 
and  has  already 
been  made  the  sub- 
ject of  many  inves- 
tigations.  Its 
chief  applications 
'hitherto  have  been 
in  surgery,  where 
it  enables  the  sur- 
geon to  locate  for- 
eign bodies  in  the 
flesh  without  prob- 
ing, and  to  know  the 
exact  nature  and  FIG-  20°- 

extent  of  fractures  and  abnormal  growths  of  the  bones. 

Physicians  are  now  able  to  examine  the  lungs  and 
other  internal  organs  with  great  ease,  and  without  dis- 
comfort to  the  patient. 


162.  The   Role   of  Wave   Motion.  —  We  are   now 


260  LIGHT. 

prepared  to  appreciate  the  important  role  which  wave 
motion  plays  in  our  lives.  By  it  we  gain  most  of  our 
knowledge  of  the  world  outside  us,  which  conies  to  us 
through  the  eye  and  ear.  Practically  all  of  the  energy 
which  keeps  our  world  alive  and  warm,  the  seat  of 
ceaseless  change,  the  home  of  myriad  living  plants  and 
animals  and  men,  instead  of  a  dark,  cold  clod,  conies  to 
us  from  the  sun,  across  the  silent  spaces,  in  waves  of 
the  intangible  ether. 

Exercises. 

117.  Why  are  the  shadows  cast  by  an  arc  lamp  much  sharper 
in  outline  than  those  cast  by  the  sun  ? 

118.  What  is  the  height  of  a  tree  which  at  a  distance  of  40 
feet  from  the  opening  of  a  pinhole  camera,  gives  an  image  4 
inches  high  on  the  ground  glass  of  the  camera  8  inches  back 
of  the  opening? 

119.  When  the  sun  is   45°  above  the  horizon,  how  long  is 
the  shadow  cast  by  a  pole  90  feet  high  ? 

120.  (a)  Construct  for  the  image  of  an  object  placed  8  cm. 
in  front   of  a  concave  mirror  of  24  cm.    radius.     (6)  Is   the 
image  real  or  virtual,  (c)  upright  or  inverted,  ((7)  enlarged  or 
diminished  ? 

121.  Construct  for  the  image  of  an  object  8  cm.  high  placed 
18  cm.  in  front  of  the  mirror  used  in  Ex.   120,  and  answer 
questions  6,  c,  cZ,  in  regard  to  it. 

122.  Construct  for  the  image  of  an  object  8  cm.  high  placed 
32  cm.  in  front  of  the  same  mirror,  and  answer  the  same  ques- 
tions in  regard  to  it. 

123.  (a)  Construct  for  the  image  of  an  object  12  cm.  high 
placed  10  cm.  in  front  of  a  convex  mirror  having  a  radius  of 


EXERCISES.  261 

30  cm,     (6)  Is  the  image  real   or  virtual,  (c)  upright  or  in- 
verted, (cZ)  enlarged  or  diminished  ? 

124.  Observe  the  image  of  your  face  in  the  convex  side  of  a 
bright  tablespoon,  and  explain  the  difference  in  the  images  de- 
pending upon  whether  the  spoon  is  held  vertically  or  horizon- 
tally: 

125.  What  must  be  the 'height  of  a  plane  mirror  in  order 
that  a  person  may  see  his  whole  length  in  it  at  orfce  ? 

126.  (a)  Does  an  Indian  spearing  fish  have  to  apply  some 
knowledge  of  the  laws  of  refraction  ?     (6)  Does  he  aim  above 
or  below  the  apparent  position  of  the  fish  ? 

127.  Where  must  the  object  be  placed  with  reference  to  a 
convex  lens  if  the  image  is  to  be  (a)   enlarged  and  real,  (6) 
enlarged  and  virtual,   (c)  inverted  and   enlarged,   (d)  upright 
and  enlarged,  (e)  real  and  of  the  same  size  as  the  object? 

128.  (a)  In  what  respects  do  all  images  formed  by  concave 
(diverging)    lenses   agree?     (6)    In  what   respects   may   they 
differ  ? 

129.  To  what  are  the  colors  due  in  (a)  ice  which  has  been 
shattered  by  a  blow,  (6)  deep  sea  water,  (c)   shoal  sea  water, 
(d)  rings  about  the  moon,  (e)  the  sunset  sky? 

130.  Consult  the  index  under  "  radiant  energy,"  look  up  all 
the  references  and  write  a  summary  of  all  you  have  learned  on 
that  subject. 


PART   II. 


LABORATORY   EXERCISES, 


PART   II.  — LABORATORY   EXERCISES. 


INTRODUCTION. 

Physics  is  to-day  the  most  exact  of  the  sciences  for 
the  reason  that  the  phenomena  with  which  it  deals  are 
capahle  of  accurate  measurement.  No  one  can  form  a 
just  conception  of  the  methods  by  which  the  great 
laws  of  physics  have  been  developed  who  has  not  him- 
self learned  to  make  at  least  some  simple  measure- 
ments with  a  fair  degree  of  accuracy.  On  the  other 
hand,  the  student  who  has  patiently  and  conscientiously 
performed  a  series  of  measurements  with  the  highest 
degree  of  accuracy  attainable  by  him  with  the  instru- 
ments at  his  command  has  gained,  not  only  manual  and 
mental  training,  but  moral  training  besides,  for  he  has 
learned  to  prefer  truth  to  falsehood  and  that  exact 
statement  of  facts  which  leads  ever  to  the  discovery  of 
truth,  to  the  loose  and  ambiguous  statement  which 
tends  to  conceal  the  truth. 

Measurement  consists  in  obtaining  an  expression  for 
the  magnitude  of  the  quantity  to  be  measured  in  terms 
of  some  standard  quantity  called  a  unit. 

The  Fundamental  Units.  All  physical  quantities 
may  be  expressed  directly  or  indirectly  in  terms  of 
some  one  or  more  of  the  fundamental  quantities, 
length,  mass,  time,  the  units  of  which  are,  respectively, 
the  centimetre,  the  gram,  and  the  second. 


266  INTRODUCTION. 

While  the  instruments  by  means  of  which  the  differ- 
ent quantities  are  measured  may  seem  at  first  sight 
quite  unlike  each  other,  it  is  yet  true  that  the  actual 
observations  made  in  measuring  these  quantities  are 
very  much  alike.  When  the  instrument  has  been  ad- 
justed for  the  observation,  the  observation  itself  Avill 
usually  consist  in  reading  or  counting  a  number  of 
linear  divisions  on  a  scale  of  equal  parts  and  the  esti- 
mation of  a  fraction  of  a  division  by  means  of  the 
eye. 

The  student  is  urged,  therefore,  to  observe  in  the 
performance  of  each  exercise,  not  only  the  precautions 
there  mentioned  for  securing  accuracy,  but,  as  far  as 
they  apply,  the  precautions  observed  in  the  preceding 
exercises.  A  little  time  spent  in  considering  before- 
hand the  probable  sources  of  error  in  any  problem  may 
save  time  in  the  end. 

The  various  sources  of  error  to  which  observers  are 
liable  will  be  mentioned  in  the  exercises.  Accidental 
errors  may  be  eliminated  by  making  a  number  of  inde- 
pendent observations  and  taking  their  mean,  since  it  is 
probable  that  as  many  observations  will  give  a  result 
above  the  true  value  as  below  it.  It  is  to  be  borne  in 
mind  that  the  mean  of  ten  observations  is,  as  a  rule, 
entitled  to  ten  times  as  much  weight  as  is  any  one  of 
the  ten,  even  though  that  one  be  exactly  the  same  as 
the  mean.  The  student  will  understand  without  spe- 
cific directions  that  every  measurement  which  he  is 
asked  to  make  is  to  be  repeated  at  least  twice,  often 


IN  TE  OD  UC  TIOX.  267 

ten  or  more  times,  and  that  all  the  observations  are  to 
be  recorded,  even  if  they  should  happen  to  be  exactly 
alike. 

If  some  observations  differ  much  more  from  the 
mean  than  most  of  those  taken,  try  to  find  a  cause  for 
the  difference  in  the  conditions  surrounding  the  experi- 
ment, and,  if  possible,  remove  the  cause  of  error. 

The  form  in  which  the  record  of  work  is  to  be  kept 
may  be  left  to  the  taste  of  the  instructor.  All  instruc- 
tors will  probably  agree  that  the  substance  of  the 
report  should  include  (1)  a  statement  of  the  object 
and  the  method  of  the  exercise;  (2)  a  description  of 
the  apparatus  used,  with  such  sketches  as  are  needed 
to  make  the  description  clear;  (3)  an  orderly  record  of 
the  observations  made;  (4)  a  discussion  of  the  results 
and  method. 

A  Personal  Hint.  Not  the  least  of  the  advantages 
to  be  derived  from  a  course  in  laboratory  work  is  the 
habit  of  order  and  neatness.  We  owe  it  not  only  to 
the  people  who  are  our  laboratory  mates,  but  to  the 
people  who  are  to  be  near  us  in  after  life,  to  see  to  it 
that  the  place  which  we  have  occupied  is  at  least  as 
clean  and  tidy  when  we  leave  it  as  when  we  came  to  it. 
If  the  teacher's  directions  in  regard  to  the  care  of  ap- 
paratus are  explicitly  followed  by  all  the  members  of  a 
class,  the  added  pleasure  derived  from  the  work  will 
alone  more  than  repay  the  effort. 


CHAPTER   IX. 
LENGTH. 

163.  Units  of  Length.  —  The  metre  is  divided  into 
100  centimetres  exactly  as  the  dollar  is  divided  into 
100  cents.  The  centimetre  is  divided  into  10  milli- 
metres as  the  cent  is  divided  into  10  mills.  All  lengths 
may  be  expressed  in  centimetres  and  decimals  of  a 
centimetre,  and  are  understood  to  be  so  expressed  in  all 
computations  unless  the  contrary  is  stated. 

Small  dimensions  when  expressed  accurately  are 
often  mentioned  in  millimetres.  Thus  an  inch  is  said 
to  be  equal  to  about  two  and  a  half  centimetres,  or  more 
exactly  25.4  millimetres. 

1.     TABLE  OF  EQUIVALENTS. 

Length. 

1  centimetre  =  .3937  in. 
1  metre  =  39.37  in.  —  3.28  ft.  =  1.0936  yd. 
1  kilometre  =  0.6214  mile. 

1  inch  =  2. 54  cm. 
1  foot  =0.3048  m. 
1  yard  =  0.9144m. 
1  mile  =  1.0094  km. 

Area. 

1  sq.  cm.  =  0.155  sq.  in. 
1  sq.  m.  =  10.714  sq.  ft.  =  1.196' sq.  yd. 


ESTIMATION   OF   TENTHS. 


9 


2<; 

1  hectare  =  2. 471  acres. 
1  sq.  km.  =  0.386  sq.  mile. 

1  sq.  in.  =  6.452  sq.  cm. 
1  sq.  ft.  —  929.034  sq.  cm. 
1  sq.  yd.  —  0.83613  sq.  m. 
1  acre  =  0.4047  hectare. 
1  sq.  m.  =  2.59  sq.  km. 

Volume. 

1  cu.  cm.  =  0.061  cu.  in. 
1  cu.  m.,  or  stere  =  35.315  cu.  ft. 
1  litre  =  1. 0567  qt.  (U.  S.). 

1  cu.  in.  =  16.387  cu.  cm. 
1  cu.  ft.  =  0.028  cu.  m. 
1  cu.  yd.  =  0.765  cu.  m. 

164.  Estimation  of  Tenths.  —  Since  the  smallest 
division  of  the  common  metre  stick  is  the  millimetre, 
we  can  measure  with  it  directly  to  millimetres.  We 
should  always  try,  however,  to  estimate  fractions  of  a 
millimetre.  We  may  at  first  find  it  more  natural  to 
estimate  fourths ;  but  since  it  is  frequently  possible  to 
estimate  closer  than  fourths,  it  is  better  to  form  the 
habit  at  once  of  estimating  tenths.  If  the  length  is  a 
certain  number  of  millimetres  plus  a  little  more  than 
i  call  the  fraction  .6,  if  a  little  less  than  f  we  may  call 
it  .7,  and  so  forth. 

EXERCISE  131. — To  measure  the  length  and  breadth  of  a 
laboratory  table. 

Apparatus.  —  Metre  stick,  a  square  block  of  wood  or  iron, 
try-square . 

Directions.  —  To    measure    the    width    of    the    table, 


270 


LENGTH. 


place  the  square  as  shown  in  Fig.  201,  at  the  left,  with 
the  ruler  resting  firmly  against  it.     The  ruler  is  then  at 
right  angles  to  the  side  of 
the  table,  and  the  end  of 
the  ruler  is  exactly  over 


FIG.  201. 


FIG.  202. 


the  edge  of  the  table.  Let 
your  assistant  hold  the 
ruler  in  place  while  you  remove  the  square,  place  it  as 
shown  at  the  right-hand  side  of  Fig.  201,  and  read  on 
the  metre  scale  the  centimetres,  millimetres,  and  tenths  of 

a  millimetre 
at  the  point 
exactly  oppo- 
site the  edge 
of  the  square. 
Make  s  u  c  h 

measurements  at  intervals  along  the  length  of  the  table. 
To  measure  the  length,  place^  the  metre  stick  against 
the  square  as  in  Fig.  202.  Since  the  table  is  probably 
more  than  a  metre  long,  place  the  square  against  the  other 
end  of 
the  stick. 
Hold  the 
square 
in  place 
while  you 

remove  the  stick,  and  put  in  its  place  against  the  square, 
a  square  block.  Now  hold  the  block  in  place,  remove 
the  square,  and  measure  from  the  block  (see  Fig.  203). 


FIG.  203. 


EXERCISES.  271 

Do  not  mark  the  table,  as  this  will  not  only  disfigure  the 
table,  but  prejudice  the  next  person  who  measures  it. 
This  remark  holds  good  in  general :  no  marks  are  to  be 
made  upon  apparatus  except  under  the  teacher's  directions. 

Record  your  results  without  reporting  them  to  your 
assistant,  then  let  him  make  a  similar  series  of  measure- 
ments while  you  assist.  After  you  are  both  through 
compare  results.  If  either  has  made  a  serious  blunder, 
the  comparison  will  show  it. 

It  is  well  to  record  both  your  own  results  and  those 
of  your  assistant,  keeping  them  apart,  however,  and 
designating  them  by  name. 

Note  the  suggestions  in  regard  to  the  record  of  results 
on  page  267.  The  measurements  may  be  tabulated  in 
some  such  form  as  the  following : 

DIMENSIONS  OF  TABLE  No.   4. 
First  breadth  at  East  end.        First  length  at  North  side. 

Breadth  in  Cm. 

Averages.  Observer. 

Self. 


8G.28G  „ 

Stephens. 


10  86.31  86.292        86.289 


Obs.  No. 

Breadth. 

1 

86.27 

2 

86.28 

3 

86.29 

4 

86.29 

0 

86.30 

6 

86.26 

7 

86.28 

8 

86.30 

9 

86.31 

272 


LENGTH. 


The  differences  in  your  own  results  will  indicate  differ- 
ences in  the  dimensions  of  the  table  at  different  points, 
provided  that  your  assistant's  results  show  corresponding 

differences. 

Length  in  Cm. 

Observer. 
Self. 


Obs.  No. 

Length. 

1 

186.26 

2 

186.27 

3 

186.26 

4 

1.86.25 

5 

186.26 

6 

186.26 

186.263 


Stephens. 


186.257        186.260  „ 

EXERCISE  132.  — To  measure  the  height  of  a  column  of  liquid 
in  a  glass  tube. 

Apparatus.  —  Metre  stick,  two  blocks,  flat  stick  or  strip  of 
plate  glass. 

Directions.  —  Place  a  flat  stick 

in  the  horizontal  position  across 
two  blocks  as  near  the  tube  as 
possible,  and  measure  from  the 
top  of  the  stick  (Fig.  204)  to  the 
upper  liquid  surface,  and  then 
from  the  same  level  to  the  lower 
liquid  surface.  The  sum  (or 
difference)  of  these  two  meas- 
ures is  the  height  required. 

Care  must  be  taken  in  making 
measurements  (a)  to  define  the 
surface  :  that  is  considered  to  be 
the  lowest  part  of  the  surface  of 
FIG.  204.  a  %uid  which,  like  water,  wets 


EXEBCISES.  273 

the  glass,  the  highest  part  of  a  mercury  surface  ;  (b)  to  see 
that  the  metre  stick  is  held  in  the  vertical  position  ;  (c)  to 
make  sure  that  the  eye  is  in  the  same  horizontal  plane  with 
the  surface  to  be  measured.  A  good  way  to  do  this  is  to 
hold  a  bit  of  mirror  or  a  piece  of  bright  tin  against  the 
back  side  of  the  metre  stick  and  slide  a  try-square  along 
the  front  of  the  metre  stick  till  the  edge  of  the  square,  its 
reflection  in  the  mirror,  and  the  surface  are  all  in  line. 

The  error  due  to  the  eye's  being  held  in  any  position 
not  in  a  perpendicular  line  to  the  scale  at  the  point  of 
observation  is  called  parallax.  It  will  be  necessary  to 
guard  against  parallax  in  almost  every  measurement 
you  will  ever  make.  Parallax  is  greater  the  farther  the 
object  is  away  from  the  scale. 

After  making  two  careful  measurements  change  the 
height  of  your  base  line,  by  turning  the  blocks  or  using 
a  stick  of  different  thickness,  and  make  two  more. 

To  measure  downward  from  the  block  to  the  lower 
surface  a  piece  of  metric  ruler  having  a  point  projecting 
from  the  end  should  be  used.  For  acids  the  point  should 
be  of  glass.  If  the  ruler  is  sawed  off  an  amount  equal 
to  the  length  of  the  point,  the  reading  on  the  ruler  will 
be  the  length  measured.  Otherwise  the  distance  from 
the  point  to  some  point  on  the  ruler  must  be  measured 
by  holding  the  piece  of  ruler  beside  the  metre  stick  while 
the  point  and  the  end  of  the  metre  stick  rest  upon  a 
piece  of  glass  or  other  hard  smooth  surface. 

EXERCISE  133.  —  To  measure  the  distance  between  two  points 
with  the  diagonal  scale. 


274 


LENGTH. 


Apparatus.  —  Dividers,  diagonal  scale,  For  the  dividers, 
any  dividers  or  compasses  with  sharp  points  will  answer. 

The  diagonal  scale  is  shown  in  Fig.  205.  It  is  designed 
to  aid  in  measuring  short  distances  more  accurately  than  can 
be  done  with  the  simple  scale  by  means  of  a  device  for  read- 
ing tenths.  The  scale  is  ruled  in  centimetres  toward  the  left 
from  zero.  To  the  right  of  zero  is  a  single  centimetre  space 
divided  into  tenths  by  diagonal  lines,  the  bottom  of  each  line 


FIG.  205. 


being  just  1  millimetre  to  the  right  of  the  top  end  of  the  line. 
The  space  between  the  top  and  bottom  of  the  scale  is  divided 
by  horizontal  lines  into  ten  equal  parts. 


EXEECISES.  27  ~) 

t 

The  first  horizontal  line  must  intersect  the  first,  or  zero, 
diagonal  line  .1  millimetre  to  the  right  of  the  nearest  straight 
line,  which  is  the  zero  of  the  scale.  The  intersection  of  hori- 
zontal 5  with  diagonal  4  is  4.5  mm.  to  the  right  of  zero  of  the 
scale. 

Directions.  —  Adjust  the  dividers  so  that  the  hinge 
does  not  work  too  easily  and  spread  the  points  so  that 
when  one  is  placed  upon  the  centre  of  one  of  the  given 
points  the  other  will  exactly  reach  to  the  centre  of  the 
second  point.  Now  place  the  dividers  upon  the  scale 
with  the  left  leg  upon  that  centimetre  division  which 
will  bring  the  right  leg  within  the  diagonal  square. 
Starting  with  the  dividers  at  the  top  of  the  scale  move 
downward,  keeping  the  left  leg  on  the  vertical  line  and 
both  legs  always  in  the  same  horizontal  line  till  the 
right  leg  meets  the  intersection  of  a  diagonal  line  with 
a  horizontal.  The  number  of  the  centimetre  line  will 
give  the  centimetres,  the  number  of  the  diagonal  line 
the  millimetres,  and  the  number  of  the  horizontal  line 
the  tenths  of  a  millimetre.  The  reading  in  Fig.  205  is 
3.26  cm. 

CAUTIOX.  —  Do  not  press  upon  the  scale  with  the  divid- 
ers, else  it  will  soon  become  blurred  and  useless. 

After  making  a  measurement,  close  the  dividers  and 
repeat  the  measurement. 

EXERCISE  134. — To  test  your  ability  to  estimate  tenths  of 
a  scale  division. 

Apparatus. — Dividers,  diagonal  scale,  metre  stick. 

Directions.  —  Make  three  measurements  of  the  dis- 
tance between  two  points  not  before  measured,  using 


276  LENGTH. 

the  dividers  and  metre  stick,  and  estimating  tenths  of  a 
millimetre  as  Avell  as  you  can  with  the  eye,  then  make 
three  measurements,  using  the  diagonal  scale,  and  com- 
pare your  results.  Try  not  to  let  any  of  your  measure- 
ments be  influenced  by  those  already  made.  If  any  of 
a  set  of  measurements  is  better  than  the  rest  it  ought 
to  be  the  last  one  rather  than  any  of  the  others. 

EXERCISE  135. — To  measure  the  diameter  of  a  sphere  (a) 
with  the  calipers,  (b)  with  the  metre  stick. 

Apparatus.  —  Outside  calipers,  two  rectangular  blocks,  flat 
stick,  metre  stick,  heavy  hammer  handle. 

The  calipers  shown  in  Fig.  206  are  called  outside  calipers,  and 
are  used  in  measuring  the  diameters  of  spheres  and  cylinders. 
Directions.  —  (a)  Indicate  different  diameters  on  the 
sphere  with  chalk.     Open  the   calipers  a  little   wider 

than  what  you  judge  to  be  the 
diameter  of  the  ball  and  make  a 
trial.  If  they  are  open  a  little 
too  wide  take  the  calipers  loosely 
in  the  hand  with  the  hinge  be- 
tween the  thumb  and  forefinger, 
and  tap  gently  with  one  leg  of 
the  calipers  upon  the  hammer 
handle  or  any  block  that  will 
neither  mar  the  calipers  nor  be 
FIG-206-  damaged  by  them.  A  little 

practice  will  enable  you  to  judge  how  hard  a  blow  to 
strike.  If  the  legs  are  a  little  too  close  together  slip  the 
hammer  handle  between  them  and  give  a  slight  blow 
upon  it  with  the  inside  of  the  caliper  leg. 


EXERCISES. 


277 


When  the  sphere  will  exactly  slip  between  the  point's, 
measure  the  distance  the  two  points  are  apart  with  the 
metre  stick  or  diagonal  scale,  open  the  calipers  and 
then  measure  another  diameter.  Measure  six  diameters. 


FIG.  207. 

Place  the  blocks  upon  the  table  against  the 
straight  stick  and  notice  whether  they  fit  squarely 
against  each  other  (see  Fig.  207).  If  all  sides  are  not 
equally  square  choose  the  best  sides,  place  the  sphere 

between  the  blocks  and 
press  the  blocks  against 
it.  See  that  the  blocks 
are  pressed  firmly 
against  the  stick. 
Now  measure  the  distance  between  the  blocks  on  both 
sides,  near  the  top  and  near  the  bottom.  Repeat  the  meas- 
urement for  three  diameters  at  right  angles  to  each  other. 

EXERCISE  136.  —  To  measure  the  inside  and  outside  diameters 
of  a  tube  with  the  calipers. 

Apparatus.  —  Outside  calipers,  inside  calipers  (Fig.  208), 
diagonal  scale. 


FIG.  208. 


278  LENGTH. 

Directions.  —  Proceed  as  in  Exercise  135  (a),  making 
four  measurements  at  each  end  for  the  inside  diameter, 
two  near  the  end  at  right  angles  to  each  other,  two  as 
far  in  as  the  calipers  will  go. 

Make  eight  measurements  of  the  outside  diameter 
along  one  element  and  eight  more  along  the  element  at 
right  angles  to  the  first.  Record  the  first  eight  results 
in  one  column  and  the  second  eight  in  another,  and 
compare  the  values  to  see  if  there  is  any  indication  that 
the  tube  is  flattened.  Glass  tubes  are  not  unlikely  to 
be  thus  slightly  elliptical  in  cross  section. 

With  the  outside  calipers  measure  the  thickness  of 
the  walls  of  the  tube,  and  see  how  it  compares  with  the 
thickness  obtained  by  taking  half  the  difference  between 
the  outer  and  inner  diameters  of  the  tube. 

165.  The  Vernier. — The  vernier  is  a  device  for 
estimating  tenths  of  a  scale  division  so  simple  in  prin- 
ciple and  construction  that  it  may  be  applied  to  a  great 
variety  of  instruments. 

The  vernier  consists  of  a  movable  scale  arranged  to 
slide  along  a  fixed  scale.  For  simplicity  let  us  describe 
a  vernier  which  reads  tenths.  When  the  zero  of  the 
vernier  (Fig.  209)  is  opposite  the  zero  of  the  scale 
the  tenth  division  of  the  vernier  will  be  found  to  be 
opposite  the  ninth  division  of  the  scale.  Each  division 
of  the  vernier  is  just  .9  of  a  scale  division  in  length. 
If  then  the  vernier  be  set  so  that  1  of  the  vernier  is 
opposite  1  of  the  scale,  0  of  the  vernier  is  .9  to  the 


THE    VEENIEE. 


279 


left  of  1  or  .1  to  the  right  of  0.  But  since  the  zero  of 
the  vernier  is  the  point  to  be  read,  the  reading  is  .1 
scale  division.  The  reading  on  the  scale  shown  in 
Fig.  209  is  12.3. 

For  the  0  of  the  vernier  is  past  12  of  the  scale  and 
3  of  the  vernier  is  opposite  15  of  the  scale,  so  that  the 
0  of  the  vernier  is  3  X  .9  =  2.7  to  the  left  of  15,  or 
the  reading  is  15  —  2.7. 

In  practice  we  need  not  notice  what  division  of  the 
scale  the  vernier  is  opposite.  We  do  observe  what 
division  of  the  scale  the  zero  of  the  vernier  is  past  and 


1    1    1   I   1    .... 

/                       \ 

1    1    1    1    1    1    1    1 

r  1  1  1  1  1  1  II 

-*.     I    I    I    I 


|    1    i    I    1 


\ 


FIG.  209. 

what  division  of  the  vernier  coincides  with  a  division 
of  the  scale. 

Verniers  are  often  made  to  read  twentieths  of  a 
millimetre,  fiftieths  of  an  inch,  sixtieths  of  a  degree,  etc. 

Before  using  a  vernier  set  it  so  that  its  zero  coincides 
with  a  division  of  the  scale,  follow  along  the  vernier 
till  the  next  division  of  the  vernier  which  coincides 


280 


LENGTH. 


with  a  division  of  the  scale  is  found,  and  count  the 
divisions  on  the  vernier  from  the  first  coincident  line  to 
the  second.  If  the  number  is  n  divisions  the  vernier 
reads  nths  of  a  scale  division. 

EXERCISE  137.  — To  measure  the  length  of  an  object  with  the 
simple  vernier. 

Apparatus.  — Metre  stick  and  vernier,  block. 

Directions. — Place  the  object  and  metre  stick  side  by 
side  against  the  block  and  slide  the  vernier  along  till  it 
touches  the  end  of  the  object  (see  Fig.  209).  Now 
read  the  vernier  as  already  directed. 

EXERCISE  138.  —  To  measure  the  diameter  and  length  of  a 
small  cylinder  with  the  vernier  gauge. 

Apparatus.  —  Vernier  gauge. 

The  vernier  gauge  (see  Fig.  210)  consists  of  a  steel  scale 
provided  with  two  jaws  at  right  angles  to  its  length.  One  of 


r 


FIG.  210. 


the  jaws  is  fixed  to  the  scale;  the  other,  to  which  a  vernier  is 
attached,  slides  along  the  scale.  When  the  jaws  are  shut  the 
zero  of  the  vernier  should  coincide  with  the  zero  of  the  scale, 
for  the  distance  apart  of  the  jaws  is  then  zero.  The  movable 
jaw  is  usually  provided  with  a  screw  which  holds  the  jaw  in 
place  while  the  reading  is  being  taken. 


THE    VEEN  IE  ft.  281 

Directions.  — Place  the  object  between  the  jaws  of  the 
gauge,  press  the  movable  jaw  gently  but  firmly  against  it, 
set  the  screw,  and  read  the  scale  and  vernier.  Make  ten 
measurements  of  the  diameter  and  four  of  the  length. 

Correction.  —  As  remarked  above,  the  reading  should 
be  zero  when  the  jaws  are  closed,  but  it  may  happen 
that  the  jaws  have  been  slightly  bent,  so  that  the  read- 
ing will  vary  one  or  two  tenths  from  zero.  Close  the 
jaws  and  read  the  vernier.  The  error,  if  any,  should  be 
recorded  as  the  error  of  zero.  It  is  evident  that  if  the 
instrument  reads  0.1  at  0  it  will  read  too  high  by  .1  at 
other  points  as  well,  and  .1  must  be  deducted  from  all 
readings,  or  what  amounts  to  the  same,  from  the  average 
of  the  readings  taken.  Bear  in  mind  that  the  correction 
is  —  .1  when  the  error  is  -f-  .1,  or  in  general  the  correc- 
tion is  the  error  with  the  sign  changed,  or  the  correction 
is  the  amount  to  be  added  to  make  the  result  correct. 

EXERCISE  139.  —  To  measure  the  internal  diameter  of  a  ring 
or  hollow  cylinder  with  the  vernier  gauge. 
Apparatus.  —  Two  vernier  gauges. 

Directions.  —  Close  the  jaws  of  the  first  gauge  and 
measure  with  the  second  gauge  the  distance  across  the 
jaws.  This  amount  must  be  added  to  all  inside  readings 
taken  with  this  particular  instrument,  since  it  is  a  nega- 
tive error. 

Record  it  thus : 

Correction  for  width  of  jaws  of  vernier  gauge  No.  4,  1.60  cm. 

Correction  for  error  of  zero,  —  0.01 


Total  correction  for  inside  measurement,  1.59  cm. 


282  LENGTH. 

The  gauge  used  for  inside  measurements  of  tubes 
or  rings  must  have  its  jaws  rounded  on  the  outside. 
The  measurements  are  to  be  taken  as  in  Exercise  138. 

166.  Eye  Estimation  of  Lengths.  —  It  is  often  use- 
ful to  be  able  to  form  a  rough  estimation  of  distances 
by  means  of  the  eye  alone,  as,  for  instance,  if  we  are 
selecting  a  block  to  support  an  object  at  a  certain  height. 
One  who  is  much  accustomed  to  such  eye  estimation 
will  detect  any  gross  error  of  measurement  by  its 
means.  Now  that  the  student  has  become  familiar  with 
the  unit  of  length  he  will  find  it  useful  exercise  to  esti- 
mate distances  habitually  before  measuring  them. 

EXERCISE  140. — To  estimate  and  measure  the  length  of 
several  objects. 

Apparatus.  —  Metre  stick. 

Directions.  —  (a)  Bend  the  forefinger  and  estimate 
the  length  from  the  tip  of  the  nail  to  the  first  joint. 
Record  your  estimate  and  then  measure  the  length. 
(5)  Spread  the  hand  and  estimate  the  distance  you  can 
span  with  the  thumb  and  middle  finger.  Record  and 
measure.  (<?)  Estimate  and  measure  the  length  of 
your  foot,  allowing,  as  well  as  you  can,  for  the  distance 
the  shoe  projects  at  each  end.  (c?)  Estimate  and  meas- 
ure the  distance  from  the  tip  of  your  middle  finger  to 
your  elbow,  (e)  Estimate  and  measure  the  length  of 
your  steps  by  taking  one  fifth  of  the  distance  covered 
in  five  steps.  (/)  Estimate  and  measure  the  height  of 
another  student. 


THE  MICROMETER   SCREW.  283 

In  olden  times  these  lengths,  when  referred  to  the 
person  of  the  king,  served  as  units  of  length  and  cor- 
responded to  (a)  the  inch,  (5)  the  span,  (<?)  the  foot, 
(d)  the  cubit,  (e)  the  yard,  (/)  the  fathom. 

The  form  of  an  object  and  its  position  influence  our 
judgment  of  its  dimensions. 

EXERCISE  141.  —  Estimation  in  different  positions. 

Directions.  —  (a)  Estimate  and  measure  the  width  and 
height  of  the  two  parts  of  the  eight  shown  in  Fig.  211. 

(5)  Estimate  and  measure  the  length  of  a  cylinder 
lying  on  its  side,  and  of  another  cylin- 
der standing  on  its  end.     A  barrel  is  a 
good  object  for  the  purpose. 

The  objects  you  have  measured  will, 
if  their  dimensions  are  borne  in  mind, 
aid  you  in  making  other  estimates  of 
length. 

167.  The  Micrometer  Screw.  —  A 

still  more  accurate  instrument  than  the 

vernier  for  measuring  small  dimensions  FIG 

is  the  micrometer  screw.     It  consists 

of  an  accurately  cut  screw,  the  head  of  which  is  divided 

into  a  number  of  equal  parts.     Its  simplest  forms  are  the 

micrometer  gauge  and  the  spherometer.* 

EXERCISE  142.  — To  measure  the  diameter  of  a  wire  with  the 
micrometer  gauge. 

Apparatus.  —  Micrometer  gauge. 

*Moro  complicated  forms  are  the  filar  micrometer,  used  in  microscopes  and 
telescopes,  and  the  dividing  engine,  used  in  ruling  accurate  scales. 


284  LENGTH. 

The  micrometer  gauge  (see   Fig.   212)   consists  of   a  bent 
arm,  one  end  of  which  is  threaded  to  receive  a   screw,   the 
other  end  containing  an  adjustable  stop,  8.     The  linear  scale 
on  the  shank  shows  how  many  millimetres  the  screw  is  from 
the  stop.     If  the  distance  between  the  threads  of  the  screw 
be  h  mm.  the  screw  will  advance  1  mm.  for  every  two  turns. 
The  head  is  divided  into  equal  parts.    If  the  threads  are  h  mm. 
apart  the  head  will  be  divided  into  50  parts.     If  the  threads 
are  1  mm.  apart  the  head  will  be  divided  into  100  equal  parts. 
In  either  case  one  division  on  the  head  corresponds  to  a 
forward  movement   of  .01  mm.  by 
the  screw.     By  estimating  tenths  of 
a  division  we  may  read  .001,  but  we 
are  to   understand   that   our   result 
cannot  be  relied  upon  as  correct  to 
.001,  since  the  error  in  setting  the 
screw  is  usually  .002  or  more. 

Directions.  —  Bring  the  screw  and  stop  in  contact 
with  a  gentle  pressure.  A  uniform  pressure  may  be 
secured  by  grasping  the  head  of  the  screw  loosely  in 
the  fingers  and  turning  till  the  fingers  slip.  Note  and 
record  the  zero  error.  Place  the  wire  against  the  stop 
and  bring  the  screw  up  to  it  with  about  the  same  pres- 
sure used  in  determining  zero. 

If  the  threads  are  £  mm.  be  careful  to  notice  whether 
the  reading  on  the  scale  is  a  whole  number  of  millimetres 
or  a  whole  number  plus  a  half.  The  reading  on  the  instru- 
ment shown  in  Fig.  212  is  2.068  mm.  If  the  screw  were 
turned  one  revolution  backward  it  would  be  2.568  mm. 
Take  ten  readings  of  the  diameter  of  the  wire,  average 
the  results,  and  determine  the  gauge  number  from 
Table  2.  Determine  the  gauge  number  also  by  means  of 
an  American  wire  gauge  such  as  is  shown  in  Fig.  213. 


THE   8PHEEOMETEE. 


285 


2.    VALUE   IN   MILLIMETRES  OF   BROWN  &   SHARP   WIRE 


GAUGE  No.'s. 

No. 

mm. 

No. 

mm. 

No. 

mm. 

No. 

mm. 

1 

7.348 

9 

2.906 

17 

1.150 

25 

0.455 

2 

6.544 

10 

2.582 

18 

1.024 

26 

0.405 

3 

5.827 

11 

2.305 

19 

0.912 

27 

0.361 

4 

5.189 

12 

2.053 

20 

0.812 

28 

0.321 

5 

4.621 

13 

1.828 

21 

0.723 

29 

0.286 

6 

4.115 

14 

1.628 

22 

0.644 

30 

0.255 

7 

3.656 

15 

1.459 

23 

0.573 

31 

0.227 

8 

3.264 

16 

1.291 

24 

0.511 

32 

0.202 

168.  The  Spherometer.  --  The  spherometer  is  a  mi- 
crometer screw  of  a  form  especially  adapted  to  the  meas- 
urement of  the  curvature  of 
spherical  surfaces.  It  may  also 
be  used  for  measuring  the  thick- 
ness of  small  glass  plates.  It 
consists  of  a  tripod,  the  three 
pointed  legs  of  which  form  an 
equilateral  triangle  (Fig.  214). 
Through  the  centre  of  the 
tripod  passes  the  pointed 
screw,  at  the  top  of  which  the 
head  is  enlarged  to  form  a  disk  which  is  graduated  on  its 
edge.  The  whole  turns  of  the  screw  are  read  from  the 
vertical  scale,  /S',  the  fractions  of  a  turn  from  the  disk. 

EXERCISE  143.  —  To  measure  the  thickness  of  a  small  glass 
or  mica  plate  with  the  spherometer. 

Apparatus.  —  Spherometer,  glass  plate. 


FIG.  213. 


286 


LENGTH. 


Directions.  —  Place  the  instrument  upon  a  piece  of 
plate  glass  and  turn  the  screw  till  it  touches  the  glass.  If 
it  is  turned  a  little  too  far  the  tripod  will  be  lifted  so  as  to 

wobble  on  its  points.  Turn 
the  screw  back  till  the  wob- 
bling motion  ceases.  All  four 
points  are  now  in  one  plane 
and  the  reading  is  the  zero 
reading.  If  the  scale  reads 
from  the  bottom  up,  the  zero 
Avill  be  near  the  middle  of  the 
scale  to  permit  of  readings 
being  made  upon  concave  sur- 
faces. Take  five  readings  of  the 
zero  in  different  positions  upon 
the  large  plate  glass.  Having 
determined  the  zero,  raise  the 
screw  till  the  plate  to  be  measured  will  slip  under  it, 
bring  the  screw  to  contact  and  read  the  instrument. 
Repeat  four  times  on  different  parts  of  the  plate.  The 
average  of  these  five  readings  less  the  average  zero  read- 
ing is  the  thickness  of  the  plate. 

EXERCISE  144. — To  measure  the  radius  of  a  large  lens  or 
other  spherical  surface. 

Apparatus.  —  Spherometer  and  glass  plate,  dividers  and 
diagonal  scale. 

Directions.  —  Determine  the  zero  reading  as  in  Exer- 
cise 143.  Raise  the  screw,  place  the  spherometer  upon 
the  lens,  bring  the  screw  to  contact  and  take  a  reading. 


FIG.  2H. 


THE   SPHEROMETER. 


Repeat  at  five  different  places  as  far  apart  as  possible  upon 

the  surface  of  the  lens.     The  difference  between  the  aver- 

age of  these  readings  and  zero  is  the  height  (A,  Fig.  215) 

of  the  point  p   above   the  plane  which  passes  through 

the  points  of  the 

tripod.      It    re- 

mains  to    meas- 

ure   d,  the    dis- 

tance of  p  from 

each  of  the  three 

points  of  the  tri- 

pod when  all  the 

points  are  in  the 

same  plane.    To 

measure  d  place 

the    instrument 

upon  a  flat  page  of  your  notebook  and  press  the  points 

down  until  they  make  dents  in  the  paper.     Bring  the 

screw  down  till  it  also  makes  a  mark.     Measure  the  dis- 

tance of  the  central  dot  from  each  of  the  others  with  the 

dividers  and  scale. 

If  R  be  the  radius  of  the  sphere  of  which  our  lens  is 
a  segment  it  is  evident  from  geometry  that 


whence: 


from  which  equation,  by  substituting  the  values  of  d 
and  h  obtained  by  measurement,  we  obtain  R. 


CHAPTER   X. 
MASS. 

169.  Mass  Defined.  Units  of  Mass.  —  By  the 
mass  of  a  body  we  mean  the  amount  of  matter  which 
composes  it.  Since  gravity  at  any  place  on  the  earth 
acts  with  equal  force  upon  equal  masses,  we  require  for 
the  measurement  of  an  unknown  mass  (1)  a  standard 
mass,  (2)  an  instrument  for  comparing  the  force  ex- 
erted by  gravity  on  the  standard  mass  and  the  unknown 
mass.  The  standard  mass  in  the  metric  system  is  the 
gram,  which  is  defined  as  the  mass  of  a  cubic  centimetre 
of  water  at  its  temperature  of  greatest  density  (4  de- 
grees Centigrade).  For  the  measurement  of  small 
masses  sets  of  weights  are  made,  usually  of  brass,  con- 
taining masses  of  1  gram,  2  grams  (two),  5  grams,  10 
grams,  20  grams  (two),  50  grams,  100  grams,  200 
grams  (two),  500  grams.  For  coarse  weighing  weights 
of  1,000  grams,  called  kilograms,  are  made.  For  frac- 
tions of  a  gram  sets  of  centigram  weights,  usually 
made  of  sheet  platinum,  and  for  very  delicate  weighing 
milligram  weights,  often  made  of  sheet  aluminum,  are 
used. 

3.    EQUIVALENTS  IN  WEIGHT. 

1  gram  =  15.4324  grains. 

1  kilogram  =  2.2046  pounds. 


THE  BALANCE. 


289 


4  ~ 


6  - 


8  ~ 


FIG.  216. 


1  grain  =  .0648  grams, 

1  ounce  (av.)  =  28.35  grams. 

1      „       (tr.)  =  31.1  grams. 

170.  The  Spring  Balance.  —  If  a  force  be  exerted 
to  elongate  a  spiral  spring  the  elongation  will  be  pro- 
portional to  the  force.  This  principle  is  made  use  of 
in  the  spring  balance. 

The  spring  balance  in  common  use  for  coarse  weighing 
has  been  graduated  by  the  maker 
so  that  we  have  only  to  suspend 
it  by  the  ring  at  the  top,  hang 
an  unknown  mass  from  the  hook 
attached  to  the  bottom  of  the 
spring,  and  read  off  the  weight 
directly  from  the  scale  (see  Fig. 
216).  The  difference  in  the 
force  of  gravity  at  different 
places  is  too  small  to  be  detected 
by  such  a  balance  and  may  there- 
fore be  disregarded. 

A  very  delicate  form  of  spring 
balance  is  that  known  as  Jolly's 
balance,  shown  in  Fig.  217.  It 
consists  of  a  long  spiral  spring 
made  of  fine  wire,  suspended  in 
front  of  a  scale.  The  elongation 
produced  by  suspending  a  known 
weight  from  the  wire  is  read  from  the  scale. 


FIG.  217. 

From  a 

series  of  such  determinations  the  average  elongation  for 
one  gram  is  determined. 


290  MASS. 

EXERCISE  145,  —  To  weigh  a  small  object  with  Jolly's  balance. 
Apparatus. — Jolly's  balance  and  some  small  weights. 

Directions.  —  Choose  a  well-defined  point  near  the 
bottom  of  the  spring  and  read  its  position  on  the  scale 
(an  index  is  usually  attached).  To  avoid  parallax  the 
scale  is  usually  ruled  on  a  mirror.  When  the  index 
covers  its  image  in  the  mirror  read  the  position  of  the 
index  on  the  scale.  If  a  piece  of  metre  stick  is  used 
for  a  scale,  as  in  the  home-made  form  of  balance  shown 
in  Fig.  217,  the  reading  is  taken  from  the  top  of  a 
try  square  which  just  touches  the  pan.  Record  the 
zero  reading,  place  the  object  to  be  weighed  in  the  pan 
and  read  again.  Repeat  both  observations  after  shifting 
the  zero  by  adding  a  small  additional  weight  to  the  pan. 
Now  find  the  value  of  one  gram  in  scale  divisions  by 
placing  in  the  pan  that  one  of  the  small  weights  which 
will  give  a  reading  nearest  that  given  by  the  unknown 
weight.  If  five  grams  extended  the  spring  12.5  cm., 
one  gram  would  extend  it  2.5  cm.  An  object  which 
extends  the  spring  16.3  cm.  would  weigh  6.52  grams. 


I71-  The  Lever  Balance.  —  The  principle  of  mo- 
ments furnishes 
a  convenient 
method  of  com- 
paring masses. 
If  a  rigid  bar 
(Fig.  218)  is 
supported  from 
a  triangular  fulcrum  a  little  higher  than  its  centre  of 


<^i 

6 


THE   BALANCE. 


291 


gravity,  it  will  come  to  rest  in  a  horizontal  position  when 
the  sum  of  the  moments  of  the  forces  on  the  left-hand 
side  is  equal  to  the  sum  of  the  moments  on  the  right- 
hand  side.  If  the  right  arm  is  made  much  longer  than 
the  left  arm,  while  the  left  arm  is  made  enough  heavier 
so  that  the  lever  balances,  it  is  evident  that  a  single 


o 


FIG.  219. 


weight   may  be  made  to  balance  different   masses   by 
varying  the  position  of  the  weight  on  the  right  arm. 

Such  balances,  now  known  as  steelyards,  were  in  use 
by  the  ancient  Romans.  The  long  arm  was  graduated 
to  equal  parts  by  notches  into  which  a  link  supporting 


FIG.  220. 

the  weight  could  be  dropped.  From  the  short  arm  was 
suspended  a  hook  for  supporting  the  object  to  be 
weighed  (see  Fig.  219).  A  form  adapted  to  delicate 
weighing  is  shown  in  Fig.  220, 


292  MASS. 

It  is  obvious  that  when  the  two  weights  are  at  equal 
distances  from  the  fulcrum  the  weights  are  equal  when 
the  beam  comes  to  rest  in  a  horizontal  position.  When 
the  long  arm  is  n  times  as  long  as  the  short  arm  the 
weight  on  the  short  arm  is  n  times  as  great  as  the  weight 
on  the  long  arm. 

EXERCISE  146.  —  To  weigh  an  object  with  a  steelyard. 
Apparatus.  —  Common  steelyard. 

Directions.  —  Support  the  steelyard  in  the  left  hand 
by  the  hook  farthest  from  the  heavy  end  if  there  are 
but  two  hooks,  from  the  second  if  there  are  three. 
Hang  the  body  to  be  weighed  from  the  hook  near  the 
heavy  end  and  move  the  bob  till  a  notch  is  found  in 
which  the  lever  will  be  horizontal.  If  the  object  is  too 
heavy  turn  the  steelyard  over  and  use  the  other  scale. 
After  reading  the  scale  and  recording  the  weight, 
weigh  a  known  weight  and  then  measure  the  distances 
between  the  knife  edges  and  the  length  of  a  scale  divi- 
sion, and  compute,  by  the  law  of  moments,  the  weight 
of  the  bob. 

EXERCISE  147.  — To  weigh  an  object  with  a  rider  balance. 

Apparatus. — Balance  with  unequal  arms,  set  of  rider 
weights. 

Directions.  —  Test  the  zero  of  the  instrument  and  if 
necessary  adjust  it  by  means  of  the  screw.  Place  the 
object  in  the  pan  and  adjust  one  of  the  larger  riders  till 
it  will  a  little  less  than  balance  the  object  and  add 
smaller  riders  till  the  beam  is  in  equilibrium.  The  sum 
of  the  weights  balanced  by  the  different  riders  is  the 


TEE  BALANCE.  293 

weight  of  the  object.  If  in  doubt  about  the  weight  of 
the  different  riders  weigh  a  known  weight,  as  a  five- 
cent  nickel,  which,  when  new,  weighs  five  grams. 

172.  The  Lever  Balance  with  Equal  Arms.  —  The 

hand  balance  (Fig.  221)  with  equal  arms  is  probably  the 
oldest  balance  known. 
It  was  in  universal 
use  for  weighing 
money  before  the 
practice  of  govern- 
ment coinage  became 
common.  The  trian- 
g  u  1  a  r  pin  passes 
through  the  centre  of  FIG- 221- 

the  beam  and  rests  in  a  double  link  which  is  held  in  the 
hand  or  fastened  by  a  cord  to  some  convenient  support. 
Near  the  ends  of  the  beam  and  at  equal  distances  from 
the  centre  are  suspended  from  small  pins  the  pans  of 
horn  or  metal  which  hold  the  weights.  Such  balances 
are  in  common  use  to-day  by  university  students  in  the 
German  chemical  laboratories  and  are  the  best  cheap 
balances  to  be  had. 

A  more  convenient  form  of  the  lever  balance  is  the 
prescription  balance  or  beam  balance  shown  in  Fig.  222. 
It  is  identical  in  principle  with  the  hand  balance,  but 
the  beam  is  supported  on  the  top  of  a  pillar  and  a  long 
and  light  pointer  indicates  on  the  scale  small  variations 
of  the  beam  from  the  horizontal  position.  The  pillar 


294 


MASS. 


may  be  made  in  two  parts,  the  upper  being  arranged  to 
slide  within  the  lower.  In  the  lowest  position  of  the 
beam  the  pans  rest  upon  the  base.  By  pressing  a  lever 
the  beam  is  lifted  and  the  pans  are  set  free. 

EXERCISE  148. — To  weigh  an  object  with  the  common  bal- 
ance. 

Apparatus.  —  Hand  or  beam  balance,  set  of  weights. 

Directions.  —  Allow  the  beam  to  swing  and  find 
where  the  pointer  comes  to  rest.  If  this  point  is  near 


FIG.  222. 

the  centre  of  the  scale,  note  the  exact  point ;  if  it  is  not 
near  the  centre,  add  enough  fine  shot,  grains  of  sand,  or 
bits  of  paper  to  the  proper  pan  to  bring  it  near  the 
centre.  Arrest  the  beam  by  means  of  the  lever,  ?,  or 
in  the  hand  balance  by  holding  the  beam  with  the  left 


THE  BALANCE.  295 

hand.  Place  the  object  in  the  left  pan  and  put  such 
weights  in  the  right  pan  as  you  judge  will  about  balance 
the  object.  Release  the  beam  a  little  to  see  which  side 
is  heavier.  Never  allow  one  end  of  the  beam  to  go 
very  much  lower  than  the  other,  especially  when  a  load 
is  on  the  balance,  as  the  knife  edge  of  the  fulcrum  is 
likely  to  be  injured  thereby.  A  good  rule  is  :  Never  let 
the  pointer  swing  off  the  scale. 

When  the  weights  are  found  which  will  bring  the 
pointer  nearly  to  zero  let  the  beam  come  to  rest  and 
note  the  point  of  rest.  Suppose  that  there  are  in  the 
pan  4.64  grams  and  that  the  point  of  rest  is  4  divisions 
to  the  right  of  zero.  You  now  add  1  eg.  and  find  the 
point  of  rest  2  divisions  to  the  left.  One  eg.  has  moved 
the  pointer  6  divisions.  It  would  have  taken  4/6  eg. 
to  move  it  4  divisions,  that  is,  to  bring  it  to  zero.  The 
exact  weight  is  therefore  4.647  grams.  Support  the 
beam  and  remove  the  objects  and  weights,  counting  the 
weights  again  to  make  sure  you  have  made  no  mistake. 

EXERCISE  149.  —  To  test  a  balance  by  double  weighing. 

Apparatus.  —  Balance,  weights. 

Directions.  —  We  have  assumed  in  Exercise  218  that 
the  arms  of  the  balance  are  exactly  equal  in  length. 
This  is  by  no  means  certain  to  be  the  case,  especially  in 
a  cheap  balance,  nor  is  it  necessary  that  the  arms  be 
exactly  equal  in  order  that  we  may  weigh  correctly, 
provided  we  know  the  ratio  of  the  lengths. 

Weigh  an  object  as  accurately  as  possible  in  the  left- 
hand  pan.  Then  weigh  the  object  in  the  right-hand 


296  MASS. 

pan.  If  we  call  the  weight  required  to  balance  the 
object  when  it  is  in  the  left  pan  W  and  the  weight 
required  to  balance  it  when  in  the  right  Wf,  while  its 
true  weight  is  M,  it  is  evident  from  the  principle  of 
moments  that  if  R  and  L  be  the  length  of  the  right 
and  left  arms  of  the  balance  (see  Fig.  223), 


OM  wQ     Qw'  MQ 

FIG.  223. 

(a)  LM=EW 

(b)  RM  =  LW> 

Dividing  (b)  by  (a) 

(c)  E/L  =  LW'/BW 
Multiplying  both  sides  by  R/L 

(d)  R2/L*=W'/W 

'      whence     (43)          B/L  =  \/W'/W 
But  from  (a),  by  dividing  by  .L, 

(45)  M  =  ~W 

x/ 

It  follows  that  to  find  the  true  weight  with  a  balance 
whose  arms  are  not  equal  we  have  only  to  determine 
RjL  by  double  weighing  once  for  all  and  afterward 
multiply  the  apparent  weight  of  the  object  in  the  left- 
hand  pan  by  RjL. 

It  is  further  true  that  for  nearly  equal  arms 

W+W 
(46)        M=  - 


THE  BALANCE.  297 

Having  found  the  true  weight  by  double  weighing,  check 
your  result  by  the  method  of  counterpoising  described 
in  the  following  exercise. 

EXERCISE  150.  — To  weigh  by  the  method  of  counterpoising. 

Apparatus.  — Balance  and  weights,  dish  of  sand. 

Directions.  —  With  the  object  in  the  left  pan  put 
enough  sand  or  shot  in  the  right  pan  to  exactly  balance 
it.  Now  remove  the  object  and  put  weights  enough  in 
its  place  to  balance  the  sand. 

EXERCISE  151.  —  To  weigh  an  object  by  the  method  of  swings. 
Apparatus.  —  Balance  and  weights. 

Directions.  —  Find  the  zero  of  the  balance.  In  a  deli- 
cate balance  the  beam  does  not  come  to  rest  for  some 
time,  but  the  point  of  rest  may  be  determined  by  ob- 
serving the  extreme  position  of  the  pointer  for  three 
successive  swings.  Since  the  am- 
plitude of  swing  is  gradually  grow- 
ing less,  owing  to  the  resistance  of 
the  air,  the  average  of  the  first  and  '"/""(<  "M"' 

second    swings    would  give    us   a  I 

point  a  little  to  the  right  of  zero,  2  I 

if  the  first  swing  was  toward  the        /'//<•, /iMM,,m»'l 
right,   while    the    average    of    the 
second  and  third  would  give  us  a  | 

point  a  little  to  the  left  of  zero. 
Thus  in    the    example    illustrated       '""/unj.lniiHM 
in  Fig.  224  the  swings  were  *  FIG-  224- 

*  To  avoid  negative  readings  the  scale  is  numbered  from  left  to  right  usually 
in  twenty  divisions,  thus  bringing  the  zero  position  at  the  10th  division. 


298  MASS. 

14+7 


=  10.5 

>=10 


Hence  to  find  the  zero,  average  the  right-hand  read- 
ings and  the  left-hand  readings  separately  and  take  the 
mean  of  the  two  averages. 


Thus:     -  -  -  =  10 
2 

After  finding  the  point  of  rest  arrest  the  beam,  re- 
lease it  again  and  take  a  new  set  of  readings.  If  the 
two  sets  give  results  agreeing  to  a  tenth  of  a  division 
they  may  be  averaged  and  the  mean  taken  as  the  zero. 
Otherwise  a  number  of  trials  must  be  made.  If  the 
balance  is  in  a  strong  draft  currents  of  air  may  pene- 
trate the  case.  One  object  of  repeating  observations 
several  times  is  to  find  out  whether  our  results  really 
possess  the  degree  of  accuracy  of  which  the  instru- 
ments and  -methods  used  are  capable. 

For  simplicity  the  readings  given  in  the  example  were 
all  whole  numbers.  In  practice  tenths  of  a  division 
should  always  be  estimated.  If  the  zero  falls  more 
than  one  scale  division  from  the  centre  the  instructor 
should  be  asked  to  adjust  the  balance. 

Having  determined  the  zero,  which  we  will  call  p^ 
place  the  object  in  the  left  pan  and  balance  it  with 


DENSITY.  299 

weights,  IF,  as  in  Exercise  220.  When  the  adjustment 
has  been  made  by  means  of  the  rider  to  within  1  mg. 
find  the  point  of  rest  pv  by  swings,  add  1  mg.  by  means 
of  the  rider,  and  find  another  point  of  rest  p2.  The 
weight  of  the  object  is  then 

(47)    M  =  W+Pl~^Po 

173.  Density.  —  By  the  density  of  a  substance  is 
meant  the  mass  of  unit  volume  of  that  substance.  For 
purposes  of  comparison  it  is  common  to  speak  of  relative 
densities,  taking  as  our  standard  of  comparison  the 
weight  of  unit  volume  of  water  at  its  temperature  of 
greatest  density  (4  degrees  C.). 

If  a  certain  volume  of  a  given  substance  is  found  to 
weigh  7.7  times  as  much  as  an  equal  volume  of  water, 
the  relative  density  or  specific  gravity  of  the  substance 
is  7.7. 

One  cubic  centimetre  of  water  at  4  degrees  C.  weighs 
one  gram,  hence  the  number  expressing  the  weight  of  1 
cu.  cm.  of  a  substance  expresses  also  its  relative  density 
or  specific  gravity.* 

If  the  body  has  a  volume  of  v  cu.  cm.  we  must 
divide  its  mass  m  by  v  to  find  the  mass  of  1  cu.  cm. ; 
hence,  in  general,  to  find  the  density  of  a  body  we 
divide  its  mass  by  its  volume,  and  our  fundamental 
formula  for  density  is 

(48)     d  =  m/v 

*  For  the  sake  of  brevity  we  shall  use  the  word  density  to  denote  relative 
density. 


300  MASS. 

The  mass  m  we  usually  find  by  weighing  the  body. 
The  method  of  finding  the  volume  must  be  adapted  to 
the  case  in  hand.  Density  as  just  defined  should, 
strictly  speaking,  be  distinguished  as  volume  density. 
The  mass  of  unit  length  of  a  string  or  wire,  for  ex- 
ample, is  called  its  linear  density,  while  the  mass  of 
unit  surface  of  a  sheet  of  uniform  thickness  is  called 
its  surface  density.  When  nothing  is  specified  to  the 
contrary,  volume  density  is  always  understood. 

EXERCISE  152. — To  find  (a)  the  linear  density  and  (b)  the 
volume  density  of  a  wire. 

Apparatus.  —  Metre  stick,  micrometer  gauge,  balance  and 
weights. 

Directions.  —  (a)  Measure  the  length  of  the  wire 
with  the  metre  stick  and  its  weight  with  the  balance. 

mass 

(49)     Linear  density  =  = r 

length 

(6)  Measure  the  diameter  of  the  wire  and  compute  its 
volume  from  the  formula  v  =  3.1416  r2/,  where  v  is  the 
volume,  r  the  radius,  and  I  the  length  of  the  cylindrical 
wire.     Compute  the  volume  density  from  the  formula 
(48)    d  =  m/v 

EXERCISE  153.  — To  determine  (a)  the  surface  density,  (b)  the 
volume  density  of  a  sheet  of  metal  bounded  only  by  straight  lines. 

Apparatus.  —  Metre  stick,  balance  and  weights,  micrometer 
gauge. 

Directions.  —  (#)  Divide  the  given  surface  into  tri- 
angles by  drawing  diagonals.  Measure  the  base  and 
altitude  of  each  triangle.  The  sum  of  the  areas  of  the 


DENSITY.  301 

triangles  is  the  area  of  the  surface.  On  the  other  side 
of  the  surface  draw  different  diagonals  and  make  an 
independent  determination  of  the  surface.  Average 
the  two  values.  The  surface  density  is  easily  found : 

(50}     Surface  density  — -^— 

surface 

(#)  Measure  the  thickness  and  compute  the  volume, 
which  is  the  thickness  times  the  surface.  Then  the  vol- 
ume density  is : 

(48}    d  —  m/v 

EXERCISE  154.  — To  make  a  set  of  weights. 

Apparatus.  —  Balance,  weights,  sheet  lead,  aluminum  wire, 
shears,  cutting  pliers. 

Directions.  —  For  the  larger  weights,  one  gram  and 
larger,  use  heavy  sheet  lead.  First  find  the  surface 
density  of  a  rectangular  piece  of  the  lead  and  calculate 
the  size  needed  for  a  5  gram  weight.  Cut  it  too  large 
and  then  pare  it  down  gradually  till  the  exact  weight  is 
obtained.  Be  very  careful  at  the  last  not  to  take  off 
too  much.  If  you  should  get  one  too  light  use  it  for 
making  2  gram  weights.  A  strip  4  times  as  long  as  the 
5  gram  weight  may  be  used  for  a  20  gram  and  folded 
in  4  parts.  The  weights  may  be  marked  with  steel  dies 
or  scratched  with  an  awl. 

For  fractions  of  a  gram  use  two  sizes  of  aluminum 
wire,  one  for  tenths,  one  for  hundredths.  Bend  the  2 
eg.  piece  at  right  angles  near  its  middle,  and  the  5  to  a 
pentagon  to  distinguish  them. 

For  a  set  of  rider  weights  smaller  wire  should  be 
chosen. 


302  MASS. 

EXERCISE  155.  —  To  find  the  density  of  a  regular  solid  as  a 
sphere  *  or  cylinder. 

Apparatus.  —  Micrometer  gauge  or  vernier  gauge,  balance 
and  weights. 

Directions.  —  Weigh  the  body  as  accurately  as  pos- 
sible. Measure  the  dimensions  of  the  body  carefully 
and  compute  its  volume.  The  volume  of  a  sphere  is 
3.1416  D3/6  =  .5236  D3,  where  D  is  the  diameter  of 
the  sphere.  The  volume  of  a  .cylinder  is  3.1416  r2£, 
where  r  is  the  radius  and  I  is  the  length  of  the  cylinder. 

Having  found  the  mass,  m,  in  grams  and  the  volume, 
v,  in  cubic  centimetres,  compute  the  density  by  the 

formula : 

(48)     d  =  m/v 

EXERCISE  156. — To  find  the  density  of  a  liquid  with  the 
graduate. 

Apparatus.  —  Balance  and  weights,  beaker,  graduate. 

The  cylindrical  graduate  is  a  cylindrical  glass  jar  of  nearly 
uniform  diameter,  graduated  to  cubic  centimetres  (see  Fig. 
225). 

Directions.  —  Weigh  in  a  beaker  a  convenient  quantity 
of  the  liquid,  pour  the  liquid  into  the  graduate,  and 
weigh  the  beaker  with  what  remains  in  it.  This  is  a 
better  plan  than  to  weigh  the  beaker  first,  since  some  of 
the  liquid  will  adhere  to  the  beaker. 

Read  on  the  graduate  the  volume  of  the  liquid,  re- 
membering that  while  the  liquid  rises  on  the  sides  of 
the  jar,  the  main  body  of  the  liquid  is  only  as  high  as 
the  central  part.  The  jar  should  be  placed  on  a  level 

*  The  balls  used  in  bicycle  bearings  are  almost  perfectly  spherical. 


DENSITY. 


303 


table  and  the  eye  should  be  on  a  level  with  the  top  of 
the  liquid  to  avoid  parallax.     As  before, 
(48)     d  =  m/v 

EXERCISE  157.  —  To  find  the  volume  and  density  of  an  irreg- 
ular solid,  using  the  graduate. 

Apparatus.  —  Balance  and  weights,  graduate. 

Directions. 

-  The  solid 
must  be  of 
a  size  to  go 
into  the  grad- 
uate easily. 
Weigh  the 
solid.  Fill 
the  graduate 
half  ""  full  of 
water  and 
note  the  vol- 
ume. Drop 
the  solid  into 
the  water  and 
again  note  the 
volume.  The 
increase  in 
volume  is  of 
course  the 
volume  of 
the  solid  FIG.  225. 

(see  Fig.  225).     If  the  solid  is  lighter  than  water  it 
may  be  held  below  the  surface  by  means  of  a  needle  in 


304 


MASS. 


the  end  of  a  stick.  The  volume  of  the'  needle  point 
may  be  neglected.  If  the  solid  is  a  crystal  soluble  in 
water  it  may  be  immersed  in  a  saturated  solution  of  the 
same  substance,  or  it  may  be  immersed  in  a  liquid  in 
which  it  is  not  soluble. 

EXERCISE  158. — To  find  the  volume  and  density  of    an 
irregular  solid  by  Archimedes'  principle. 

Apparatus.  —  Balance   and   weights,  tumbler,   bridge,   fine 
thread. 

Directions.  —  By  Archimedes'  principle  a  solid  when 
immersed  in  a  liquid  has  its  apparent  weight  dimin- 
ished by  an  amount  equal  to 
the  weight  of  the  liquid  dis- 
placed. If  the  liquid  be  water 
the  number  expressing  its  weight 
in  grams  expresses  also  its  vol- 
ume in  cubic  centimetres.  But 
the  volume  of  water  displaced 
is  evidently  equal  to  the  volume 
of  the  solid  which  displaces  it, 
whence  the  loss  of  weight  of  a 
solid  when  immersed  in  water  is 
the  volume  of  the  solid. 

Place  the  bridge  over  the  left- 
hand  pan  of  the  balance,  taking 
care  that  it  does  not  touch  the 
pan  (see  Fig.  226).  Set  the 
tumbler  upon  the  bridge  and  susperid  the  solid  by  a 
thread  from  the  hook  supporting  the  balance  pan.  The 


FIG.  226. 


DENSITY.  305 

body  should  hang  free  in  the  middle  of  the  glass. 
Weigh  the  body  and  then  pour  water,  which  has  been 
previously  boiled  to  free  it  from  air,  into  the  glass  till 
the  body  is  completely  covered  and  weigh  again.  The 
difference  in  weight  will  not  give  exactly  the  volume 
of  water  displaced,  since  a  gram  of  water  at  any  higher 
temperature  than  4  degrees  has  expanded  to  a  volume 
greater  than  1  cu.  cm.  Note  the  temperature,  therefore, 
and  look  up  in  Table  4  the  volume  of  one  gram  of  water 
at  the  given  temperature.  Multiply  the  loss  of  weight 
in  water  by  the  factor  found  in  the  table.  This  will  give 
you  v.  As  usual, 

(48)     d  =  m/v 

EXERCISE  159. — To  find  the  density  of  a  liquid  by  Archi- 
medes' principle. 

Apparatus.  —  As  in  Exercise  158. 

Directions.  —  Suspend  a  glass  stopper  as  directed  in 
Exercise  157.  Weigh  the  stopper  in  air  and  in  water 
and  so  find  its  volume.  Now  remove  the  water  and  fill 
the  glass  with  the  given  liquid.  A  saturated  solution 
of  salt  or  of  copper  sulphate  may  be  used.  Find  the 
loss  of  weight  of  the  stopper  in  the  given  liquid.  This 
is  equal  to  the  weight  of  a  volume  of  the  liquid  equal 
to  the  volume  of  the  stopper.  Again, 

(48)     d  =  m/v 

EXERCISE  160. — To  find  the  density  of  a  liquid  with  the 
bottle. 

Apparatus.  —  Balance  and  weights,  glass-stoppered  bottle. 


S06  MASS. 

The  specific  gravity  bottle  (Fig.  227)  furnished  by  dealers  is 
usually  a  small  Florence  flask  with  a  mark  on  its  neck  or,  much 
better,  a  light  bottle  with  a  perforated  ground-glass  stopper,  to 
facilitate  filling  the  bottle  without  leaving  air  near  the  top. 
Any  light  glass-stoppered  bottle  holding  from  20  to  50  c.c. 
will  answer  the  purpose  nearly  as  well. 

Directions.  —  Wash  the  bottle  and  dry  it  by  putting 
a  little  alcohol  in  it.  After  shaking  well  so  that  the 
water  will  be  taken  up  by  the  alcohol, 
pour  the  alcohol  into  the  flask  kept  for 
impure  alcohol.  To  remove  what  re- 
mains in  the  bottle  rinse  it  with  a  little 
ether.  The  ether  which  remains  will 
soon  evaporate,  leaving  the  bottle  dry. 
Weigh  the  bottle,  fill  it  with  boiled 
water,  note  the  temperature  and  weigh. 
The  weight  of  the  bottle  and  water  less 
^^^^  the  weight  of  the  bottle  is  equal  to  the 

FIG.  227.  volume,  v,  contained  by  the  bottle. 

Empty  and  dry  the  bottle,  fill  it  with  the  given  liquid, 
and  weigh  as  before.  The  weight  is  m  and 

(48)    d  —  m/v 

EXERCISE  161.  — To  find  the  density  of  a  solid  with  the  bottle. 
Apparatus.  —  Balance  and  weights,  bottle. 

Directions.  —  Find  the  mass,  w15  of  the  bottle  when 
filled  with  water.  Weigh  the  solid  m,  and  put  it  into  the 
bottle,  allowing  the  water  to  overflow,  wipe  the  bottle 


DENSITY. 


307 


and  weigh  again,  w2.     The  volume  of  the  solid  is  then 
equal  to  the  mass  of  water  that  overflowed,  or: 

v  •=.  ml  -j-  m  —  m2 
and 

(48)     d  =  m/v 

EXERCISE  162.  — To  find  the  density  of  a  liquid  which  does 
not  mix  with  water,  using  the  U  tube. 

Apparatus.  —  Glass  tube  one  metre  long,  bent  to  the  shape 
shown  in  Fig.  228. 

Directions.  —  Support  the  tube  in  a  vertical  position 
and  pour  enough  water  into  it  to  fill  it  about  halfway 
to  the  top.  Pour  into  one  arm  enough  of 
the  given  liquid  to  fill  one  arm  nearly  to 
the  top,  taking  care  that  there  is  enough 
water  to  fill  the  bend  of  the  tube.  Meas- 
ure the  height  of  the  liquid  in  each  arm 
above  the  level  of  the  surface  of  contact 
of  the  two  liquids.  It  is  evident  that  the 
pressure  of  the  liquid  in  both  tubes  at  the 
level  A  is  equal.  If  the  tube  is  of  uniform 
diameter  the  masses  of  the  columns  AB 
and  A  0  are  equal,  and  if  the  cross  section 
of  the  tube  were  1  sq.  cm.  the  volume  of 
the  given  liquid  would  be  AB  cu.  cm.  and 
its  weight  would  be  AC  grams.  Since 
the  ratio  of  the  heights  of  the  two  col- 
umns is  the  same  whatever  the  size  of  the 
tube  may  be,  we  may  call  AB  =  v  and  A  C  =  m,  and 
we  have 

(48)     d  = 


FIG.  228. 


308 


JL 


EXERCISE  163. — To  measure  the  density  of  a  liquid  by  the 
principle  of  the  barometer. 

Apparatus.  —  Two  tumblers,  two  glass  tubes  united  at  the 
top  by  a  T  joint,  to  the  stem  of  which  is  attached  a  piece  of 
rubber  tubing  (Fig.  229). 

Directions.  —  Fill  one  tumbler  with 
the  liquid,  the  other  with  water. 
Let  one  of  the  tubes  dip  in  each 
tumbler.  If  all  of  the  air  were  now 
removed  from  the  tubes,  the  water 
B  would  rise,  if  possible,  to  a  height  of 
about  10  metres,  due  to  the  pressure 
of  the  atmosphere.  Exhaust  enough 
of  the  air,  by  applying  the  mouth  to 
the  rubber  tube,  so  that  the  water 
will  rise  nearly  to  the  top  of  the  tube 
and  close  the  rubber  tube  with  a 
pinch  cock.  If  the  other  liquid  is 
heavier  than  water  it  will  rise  to  a 
less  height.  Measure  carefully  the 
height  of  the  liquid  column  AB  above 
the  surface  of  the  liquid  in  the  tum- 
FIG.  2-29.  bier  and  of  A'O  above  the  surface 

Then,   as   in   Exercise    162,  A'C=m, 


of   the   water. 
AB  =  v,  and 


(48)    d  = 


EXERCISE  164. — To  find  the  density  of  a  substance  lighter 
than  water  by  Archimedes'  principle. 

Directions.  —  Let  the  student  devise  a  method. 


DENSITY. 


309 


EXERCISE  165.  — To  find  the  length  of  a  coiled  wire. 

Apparatus. — Same  as  for  Exercise  158,  also  micrometer  gauge. 

Directions.  —  Find  the  volume  of  the  wire  and   its 
diameter  and  compute  its  length. 


4.  DENSITIES. 

Solids. 

Aluminum 

2.58 

Pine      . 

0.5 

Brass 

8.4 

Faraffine 

."-        .89 

Copper    . 

8.92 

Platinum 

.      21.5 

Cork 

0.24 

Porcelain 

2.2 

Glass  (common) 

2.6 

Quartz           . 

2.65 

„      (flint) 

3.5 

Sand     .      t*. 

2.8 

Gold 

19.3 

Silver    . 

.       10.53 

Ice 

J).91 

Steel     . 

7.87 

Iron  (wrought) 

7.84 

Tin        . 

7.29 

Lead 

"11.33 

Wax      . 

.  1       .96 

Oak 

0.8, 

Zinc      . 

.  :    7.16 

Liquids  (20°). 

Alcohol 

.789 

Glycerine 

^       1.26 

Carbon  bisulphide 

1.29 

Mercury 

.       13.6 

Ether               .     *  ,  . 

0  74 

Sulphuric  acid 

1.85 

Water. 

t                 D 

t 

D                   t 

D 

0°          0.99988 

10° 

0.99974             20° 

0.99827 

1    ,         0.99993 

11 

0.99965             21 

0.99806 

2            0.99997 

12 

0.99955              22 

0.9978$ 

3            0.99999 

13 

0.99943             23 

0.99762 

4            1.00000 

14 

0.99930             24 

0.99738 

5            0.99999 

15 

0.99915              25 

0.99714 

6            0.99997 

16 

0.99900             26 

0.99689 

7            0.99994 

17 

0.99884             27 

0.99662 

8            0.99988 

18 

0.99866             28  - 

0.99635 

r9            0.99982 

19 

0.99847            29 

0.99007 

CHAPTER  XI. 
TIME. 

174.  Measurement  of  Time.  —  In  measuring  time, 
as  in  measuring  length,  we  must  be  able  to  divide  the 
object  to  be  measured  into  equal  parts.  The  natural 
divisions  of  time,  the  year,  the  lunar  month,  the  day, 
have  been  subdivided  into  smaller  units  for  convenience 
in  measuring  small  intervals  of  time.  Our  smallest  unit 
of  time,  the  second,  is  ^-RTFO  of  a  mean  solar  day.  The 
period  of  a  complete  revolution  of  the  earth  upon  its 
axis  is  absolutely  the  same  from  day  to  day  and  from 
year  to  year.  If  we  are  to  make  accurate  time  meas- 
urements we  must  employ  an  instrument  that  returns 
to  a  given  starting  point  after  equal  intervals  of  time, 
or,  as  we  say,  vibrates  isochronously.  This  condition  is 
met  if  a  body  is  held  in  a  position  of  equilibrium  by 
some  force  and,  if,  when  it  is  displaced  from  that  posi- 
tion, the  force  causing  it  to  return  is  proportional  to 
the  displacement.  This  is  the  sort  of  motion  which 
produces  musical  sounds,  and  it  is  known  as  simple 
harmonic  motion.  The  vibration  of  a  tuning  fork  when 
struck,  a  guitar  string  when  picked  and  released,  a  ver- 
tical spiral  spring  with  a  weight  at  its  lower  end  when 
pulled  down  and  released,  a  vertical  wire  with  a  weight 

310 


MEASUREMENT  OF   TIME.  311 

attached  when  twisted  and  released,  a  common  swing 
pendulum  when  the  bob  is  drawn  to  one  side  and  re- 
leased, the  spiral  hairspring  of  a  watch  when  distorted 
—  all  these  are  examples  of  simple  harmonic  motion. 

EXERCISE  166.  —  To  find  the  time  of  oscillation  of  a  simple 
pendulum. 

Apparatus.  —  Clock  or  watch,  metre  stick,  small  thread,  ball. 

Directions.  —  By  time  of  oscillation  we  mean  one  half 
the  time  required  for  a  complete  vibration.  The  time 
of  vibration  of  the  seconds  pendulum  is  two  seconds. 
The  pendulum  should  be  so  supported  that  the  length 
of  the  string  is  the  same  in  all  parts 
of  its  swing.  Fig.  230  shows  a  good 

way  and  a  bad  way  of  supporting  the       _      -_-.. 

string.     Adjust  the  pendulum  to  a     -"^-^--—^ 
convenient  length,  sit  with  your  eye 
in  line  with  the  string  and  a  mark 
behind  it,  such  as  the  edge  of  the 
support.     Set  the  pendulum  swing- 
ing through  a  small  arc  and  practise 
counting  the  swings  as  it  passes  the 
mark  till  you  can  look  off  for  an  instant  at  the  watch 
without  losing  the  count.     Note  the  hour,  minute,  and 
second  of    the  passage   of   the   pendulum,   count   fifty 
swings,   and  again  take  the  time.     The  difference    be- 
tween these  two  times  divided  by  fifty  will  be  the  period 
of  oscillation.     Repeat   till   your   observations    do   not 
vary  more  than  one  or  two  seconds  in  fifty  beats.     Now 
measure  the  length  of  the  pendulum  to  the  middle  of 


312  TIME. 

the  ball  with  the  metre  stick,  taking  the  average  of  the 
lengths  to  the  top  and  bottom  of  the  ball  and  compute 
the  time  of  oscillation  by  the  law  of  the  pendulum 


(31}    t  =  8.1416  ^_ 

where  I  is  the  length  of  the  pendulum  and  g  =  9800 

Make  the  same  measurements  with  the  pendulum  a 
little  longer  and  again  with  it  a  few  centimetres  shorter. 

EXERCISE  167.  —  To  find  the  time  of  oscillation  of  a  spiral 
spring. 

Apparatus.  —  Jolly's  balance,  watch. 

Directions.  —  Put  a  small  weight  in  the  pan  and  mark 
the  position  of  the  index  on  the  scale  -by  fastening  a 
little  paper  pointer  to  the  scale  with  a  bit  of  wax. 
Draw  the  spring  downward  two  or  three  centimetres 
and  release  it.  Count  the  number  of  times,  n,  that  the 
index  passes  the  mark,  both  up  and  down,  in  one  min- 
ute. The  time  of  oscillation  is  then  t  =  60  /  n.  Count 
for  five  minutes  and  the  time  is  t  =  300  /  n',  a  result 
which  is  more  probably  correct  than  the  former. 

EXERCISE  168.  —  To  find  the  time  of  oscillation  of  a  torsional 
pendulum. 

Apparatus.  —  Torsional  pendulum,  watch  or  clock,  tele- 
scope." 

Directions.  —  Fasten  a  bit  of  paper  with  a  vertical 
mark  upon  it  on  the  side  of  the  weight  facing  you  and 
set  the  telescope  so  that  the  cross  hair  in  the  eyepiece 
will  coincide  with  the  mark  when  the  pendulum  is  at 


MEASUREMENT  OF  TIME.  313 

rest  (see  Fig.  231).  If  the  clock  beats  seconds  audibly 
watch  until  the  mark  crosses  the  hair  exactly  on  a  sec- 
ond, then  count  the  seconds  till  it  passes 
again  on  an  even  second,  while  another 
student  counts  the  number  of  beats  of  the 
pendulum.  Time  should  be  given  for  at 
least  ten  beats  to  occur.  Exchange  places 
with  your  assistant  ]  and  repeat.  Make 
at  least  five  determinations  apiece.  Add 
weights  to  the  pendulum  and  again  deter- 
mine the  time. 

EXERCISE  169. — To  determine  the  acceleration 
due  to  gravity  with  the  simple  pendulum. 

Apparatus.  —  Metre  stick  with  vernier,  simple  pendulum. 

Directions.  —  Measure  £,  the  length  of  the  pendulum, 
in  centimetres,  and  t,  its  time  of  oscillation,  in  seconds, 
with  all  possible  care  as  directed  in  Exercise  166.  If 
the  pendulum  is  long,  the  error  in  measuring  I  and  t 
will  be  less  than  with  a  short  one.  The  thread  must  be 
light  and  the  bob  should  not  be  much  more  than  2  cm. 
in  diameter.  A  metal  ball  is  better  than  a  wooden  one. 
Why  ?  The  oscillations  of  a  simple  pendulum  are  not 
strictly  isochronous  except  for  a  small  arc.  With  a 
pendulum  one  metre  long  the  ball  may  swing  5  or  6 
cm.  to  each  side  of  the  centre  without  introducing  a 
perceptible  error.  Having  determined  the  value  of  t 
and  Z,  substitute  them  in  the  formula 
(33}  •  g  =  (3.1416)*  I  ft* 

EXERCISE  170,  —  To  count  your  pulse, 


314  TIME. 

Directions.  —  Lay  your  watch  on  the  table  before  you. 
Place  the  forefinger  of  the  right  hand  on  the  left  wrist, 

as  shown  in  Fig.  232. 
When  the  location  of  the 
artery  is  found  press 
slightly  with  the  forefinger 
and,  with  your  eye  on  the 
watch,  count  the  beats 
FIG- 232-  that  occur  in  one  minute. 

Repeat  three  times.     Ask  other  students  for  the  results 
obtained  by  them. 

EXERCISE  171.  — To  count  the  respiration. 
Directions.  —  Breathe  quietly  for  a  minute  while  you 
count  the  number  of  respirations.  Try  four  times. 
Watch  a  student  who  is  writing  or  otherwise  quietly 
employed  and  count  his  respiration.  Do  the  results 
obtained  agree  as  closely  as  the  time  found  for  the  pulse 
beat? 

175.  Estimation  of  Time.  —  It  is  frequently  of  use 
to  be  able  to  estimate  intervals  of  a  few  seconds  with- 
out the  watch,  as  in  making  a  photographic  exposure. 

EXERCISE  172.  —To  estimate  a  short  interval  of  time. 
Apparatus.  —  Watch. 

Directions.  —  Sit,  watch  in  hand,  and  count  with  the 
watch  for  three  minutes  or  till  you  can  count  at  about 
the  right  rate,  then  note  the  time  and  looking  off  from 
the  watch  count  for  fifteen  seconds-and  again  look  at  the 
watch.  Record  the  result  and  repeat  five  times.  Is 


MEASUREMENT  OF  TIME.  315 

your  error  always  on  the  same  side  ?  It  is  sometimes 
helpful  to  utter  a  phrase  of  about  the  right  length  to  fill 
the  interval  between  counts.  The  words,  a  thousand 
and  one,  a  thousand  and  two,  etc.,  may  be  made  to  serve 
as  a  rough  measure  of  time. 

EXERCISE  173.  — To  find  your  rate  of  walking. 
Apparatus.  —  Watch. 

Directions.  —  Count  the  steps  you  take  in  walking 
some  distance  you  cover  daily.  Repeat  ten  times  and 
estimate  the  distance  from  your  length  of  step.  Note 
the  time  required  to  walk  the  distance  at  your  usual 
rate  and  also  at  the  fastest  rate  which  you  are  able  to 
keep  up.  Record  any  difference  in  the  direction  of  the 
wind  or  the  like  that  might  affect  your  rate. 


CHAPTER   XII. 


FORCE. 

PRESSURE    OF    FLUIDS. 

EXERCISE  174.  — To  measure  the  pressure  of  the  atmosphere 
with  the  barometer. 

Apparatus. — The  best  form  of  instrument  is  Fortin's  ba- 
rometer, shown  in  Fig.  233.  The  height  of  the  barometer  is 
the  difference  in  level  of  the  two  mercury  surfaces 
s  and  s*.  The  zero  of  the  scale  is  the  tip  of 
the  ivory  point,  which  projects  into  the  cistern. 
Hence,  that  the  scale  may  measure  the  height  of 
the  column,  the  surface  s  must  be  adjusted,  by 
means  of  the  adjusting  screw  at  the  bottom,  until 
the  ivory  point  touches  the  surface  of  the  mercury. 
The  reading  is  taken  with  the  vernier,  which 
is  moved  by  means  of  the  milled  head,  m,  till  the 
light  reflected  from  the  white  background  barely 
disappears  at  the  middle  point  of  the  surface,  s!. 
To  prevent  parallax  the  vernier  is  graduated 
upon  a  movable  tube.  The  eye  is  in  the  proper 
position  to  take  a  reading  when  the  front  lower 
edge  of  this  tube  appears  to  coincide  with  the 
back  lower  edge.  The  vernier  sometimes  differs 
from  the  one  already  used  only  in  being  divided 
to  twentieths  instead  of  tenths. 

Directions.  —  Adjust  the  mercury  in  the 
cistern  till  the  ivory  point  touches  the  sur- 
face.    One  adjustment  is  sufficient  for  the 
FIG.  233.        time  °f  the  exercise.     Take  three  readings 

816 


THE  BAROMETER. 


317 


3:00 

742.25 

3:10 

742.30 

3:20 

742.35 

3:30 

742.40 

3:40 

742.40 

3:50 

742.50 

4:00 

742.40 

rs 

AVERAGE 

742.25 

742.27 

742.30 

742.30 

742.30 

742.33 

742.40 

742.40 

742.45 

742.43 

742.55 

742.52 

742.60 

742.55 

every  ten  minutes  for  an  hour.  Record  the  time  (day, 
hour,  and  minute)  of  each  observation.  Tabulate  the 
results  and  plot  a  curve  showing  graphically  the  varia- 
tions of  the  barometer  during  the  hour.*  Suppose  your 
observations  were  as  follows  : 

READING  OF  BAROMETER,  OCTOBER  8. 
TIME  OBSERVATIONS 

742.30 
742.30 
742.35 
742.40 
742.45 
742.50 
742.52 

To  plot  these  results  in  the  form  of  a  curve  we  let 
equal  horizontal  spaces  represent  equal  periods  of  time 
and  equal  vertical  spaces  represent  equal  heights  of  the 
barometer.  We  divide  the  distance  across  a  page  of 
cross-section  paper  or  any 
paper  ruled  in  squares  into 
six  equal  parts,  as  shown  in 
Fig.  234.  Then  if  the  base 
line  represent  a  height  of 
742.25  mm.  and  the  top  line 
a  height  of  742.55  mm.,  the  various  heights  given  in 
our  table  of  results  will  fall  at  the  corresponding  heights 
on  the  paper.  If  we  divide  each  vertical  space  into  ten 
parts  the  reading  for  3 : 00  will  fall  at  2.7  divisions 

*If  the  barometer  is  not  provided  with  a  vernier  follow  the  directions  given 
in  Exercise  132,  using  the  metre  stick  and  square. 


FIG.  234. 


318  FORCE. 

Vabove  the  bottom  line,  etc.  Make  a  small  circle  at  the 
point  where  the  vertical  line  which  represents  by  its 
distance  to  the  right  the  time,  meets  the  horizontal  line 
which  represents  by  its  height  the  height  of  the  barom- 
eter. Connect  all  these  circles  by  a  smooth  curve  and 
we  have  a  graphic  or  pictorial  representation  of  the  vari- 
ations of  the  barometer  between  3  P.M.  and  4  P.M.  of 
October  8. 

EXERCISE  175. —  To  determine  the  altitude  of  your  labora- 
tory above  the  level  of  the  sea. 

Apparatus.  —  Barometer,  daily  weather  maps. 

The  weather  maps  may  be  seen  at  the  post  office,  or  on  an 
application  of  the  instructor  through  the  postmaster  they  will 
be  sent  regularly  to  the  laboratory,  where  if  daily  posted  they 
may  be  made  a  source  of  much  valuable  information  in  the 
application  of  the  principles  of  physics  to  the  solution  of  the 
problems  of  our  atmosphere.  A  good  aneroid  barometer  will 
answer  for  this  experiment  if  it  has  been  compared  with  a 
mercurial  barometer  and  is  known  to  be  correct. 

Directions.  —  Read  the  barometer  daily  as  nearly  as 
possible  at  the  hour  for  which  the  map  is  drawn. 
Locate  your  town  as  well  as  you  can  on  the  map  and 
note  the  position  of  the  two  isobars  that  pass  nearest  to 
it.  This  will  give  you  the  reading  of  the  barometer 
reduced  to  sea  level,  since  all  the  observations  of  pres- 
sure made  at  the  different  stations  are  reduced  to  the 
pressure  that  would  be  found  down  a  mine  whose  bot- 
tom was  on  a  level  with  the  ocean.  The  difference  in 
pressure  is  about  1  inch  for  930  feet.  The  difference 
between  your  reading  of  the  barometer  and  the  reading 


THE  MANOMETER. 


319 


FIG.  235. 


obtained  from  the  map,  if  both  are  expressed  in  inches/" 
will,  when  multiplied  by  930,  give  the  elevation  above 
sea  level  of  your  laboratory. 

EXERCISE  176.  —  To  measure  the  pressure 
of  the  gas  in  the  laboratory. 

Apparatus.  —  Open  manometer  and  metre 
stick  with  vernier.  The  open  manometer  or 
pressure  gauge  may  be  made  of  a  glass  tube, 
bent  as  shown  in  Fig.  235.  The  bend  of  the 
tube  is  filled  with  water  and  the  horizontal 
ann  is  attached  by  a  rubber  tube  to  the  gas 
cock. 

Directions.  —  Turn  on  the  gas  and 
measure  the  difference  in  level  of  the 
two  surfaces  A,  B.  If  this  distance  be  I  centimetres 
the  volume  of  water  supported  by  the  gas  for  a  surface 
of  1  sq.  cm.  is  I  cubic  centimetres  and 
its  weight  is  I  grams.  The  pressure  is 
therefore  I  grams  or  980  X  I  dynes  per 
sq.  cm. 

EXERCISE  177. — To  measure  the  pressure 
of  the  water  at  the  hydrant  and  to  compute  the 
head  required  to  produce  it. 

Apparatus. — Closed  manometer,  metre 
stick,  and  vernier.  The  closed  manometer 
consists  of  a  bent  tube  with  one  arm  closed. 
It  has  a  mercury  index  in  the  bend  of  the  tube 
(see  Fig.  236).  The  open  end  of  the  tube  must 
be  fastened  firmly  by  means  of  wire  to  the  rubber  tube.  Both 
the  glass  tube  and  the  rubber  tube  should  be  very  strong  to 
withstand  the  pressure  upon  them. 


-D 


-B 


--C 


-A 


FIG.  236. 


320  FORCE. 

Directions.  —  Bring  the  mercury  to  the  same  level  in 
both  arms  of  the  manometer  by  tilting  it  till  a  little 
mercury  flows  from  one  side  to  the  other.  Measure  the 
length,  CD,  of  the  enclosed  air  column  and  read  the 
barometer  pressure,  p. 

Now  turn  on  the  water  gradually  till  you  have  the 
full  pressure  and  measure  the  length,  ED. 

Call  CD  ==  v,  BD  =  v<  and  AB  =  I. 

Then  from  Boyle's  law :  vp  —  v'pf,  whence  the  pres- 
sure, jt/,  of  the  enclosed  air  is 

(48}    p'  =  vp/V 

But  p'  is  not  the  only  pressure  acting  against  the 
pressure  of  the  water.  The  weight  of  a  mercury 
column  I  cm.  long  is  also  supported  by  the  water.  The 

pressure  of  the  water  is  therefore 

* 

P  =pr  -)-  I  cm.  of  mercury 
P  —  (p'  -[-  1}  x  13.55  grams  per  sq.  cm. 
(49}     P  =  (p'  -j-  I)  X  13.55  X  980  dynes  per  sq.  cm. 

CAPILLARITY. 

176.  The  Capillary  Constant  or  coefficient  of  sur- 
face tension  of  a  liquid  is  defined  as  the  weight  of 
liquid  raised  above  its  natural  level  per  unit  of  length 
of  the  line  bounding  the  surface  of  the  liquid  raised. 
In  a  capillary  tube  of  radius  r  the  length  of  the  line 
bounding  the  surface  is  2  X  3.1416  r.  If  the  liquid 
whose  density  is  d  rises  to  a  height  h  the  volume  of  the 
liquid  raised  is  3.1416  hr2  and  its  weight  is  3.1416  hr2d. 


THE  CAPILLARY  CONSTANT. 


321 


The  capillary  constant  is,  then, 


a  =  8.1416  hr*d  /  2  X  3.1416  r  =  h  hrd 


or  in  dynes 


(50)     a  =  h  hrd  X  980 


EXERCISE  178.  —  To  find  (a)  the  radius  of  a  capillary  tube, 
(b)  the  capillary  constant  of  a  liquid. 

Apparatus.  —  Clean  mercury,  balance,  watch  glass,  crystal- 
lizing dish,  pipette,  metre  stick,  block  (Fig.  237).  This  block 
consists  of  a  little  board  with  a 
scale,  made  of  a  piece  of  metre 
stick,  attached  to  it.  Two  brass 
nails  are  driven  through  the  board 
and  project  about  2  cm.  'below  the 
top  of  the  board.  , 

Directions.  —  (#)  To  measure     » 
the  radius  of  the  capillary  tube. 

See  that  the  tube  is  perfectly 
dry,  then  draw  into  it  a  thread 
of  mercury  about  3  or  4  cm. 
long.  This  may  be  done  by  attaching  a  small  rubber 
tube  to  one  end  of  the  glass  tube  and  exhausting  the 
air.  Be  careful  not  to  draw  the  mercury  into  your 
mouth.  Pinch  the  rubber  tube  to  keep  the  mercury  in 
place  while  you  lay  the  glass  tube  upon  a  metre  stick  and 
measure  the  length,  ?,  of  the  mercury  column.  Pour 
the  mercury  out  into  a  watch  glass  and  weigh  it.  The 
density  of  mercury  at  20  degrees  is  13.55.  The  volume 
of  the  mercury  thread  is  3.1416  lrz  and  its  weight  is 
m  —  13.55  X  3.1416  Zr2,  whence 


FIG.  237. 


322  FORCE. 

(51)     r2  =  m  /  13.55  X  8.1416 1 

from  which,  by  extracting  the  square  root,  we  obtain  r. 

(6)  To  measure  A,  first  clean  the  tube,  by  pouring 
nitric  acid  through  it  and  rinsing  with  water.  Set  the 
block  (Fig.  237)  upon  the  crystallizing  dish  and  fill 
the  dish  with  water  till  the  surface  of  the  water  just 
touches  the  points  of  the  nails.  Make  the  final  adjust- 
ment with  the  pipette.  Set  the  tube  in  front  of  the 
scale  and  read  the  height  of  the  liquid  on  the  scale. 
Raise  and  lower  the  tube.  If  the  tube  is  clean  the 
height  will  not  be  altered.  Remove  the  tube,  set  the 
block  upon  the  table  and  measure  the  distance  from 
the  points  to  the  bottom  of  the  scale.  Measure  at  both 
ends  and  take  the  average.  This  distance  added  .to 
the  reading  of  the  scale  is  equal  to  h.  Substitute  your 
values  of  r  and  h  and  the  value  for  rf,  obtained  from 
Table  4,  in  the  formula,  a  =  */2  hrd  X  980. 

Note  the  temperature  of  the  liquid  and  compare  your 
result  with  that  given  in  Table  5. 

5.    CAPILLAEY  CONSTANT. 

TEMPERATURE  WATER  ALCOHOL 

10°  77.5  24.97 

15°  76.6  24.53 

20°  75.7  24.09 

25°  74.8  23.60 

ELASTICITY. 

177.  Young 's  Modulus  of  Elasticity  is  defined  as 
the  force  which  would  double  the  length  of  a  rod 


YOUNG9  S  MODULUS   OF  ELASTICITY.  323 

or  wire  of  unit  cross  section  of  the  substance  under 
consideration.  It  may  also  be  defined  as  the  ratio  of 
the  stress  to  the  strain.  If  a  wire  of  length  L  and 
cross  section  a  is  elongated  an  amount  I  by  a  weight  W 

W 

the  stress  per  unit  area  is  —  ,  while  the  strain  per  unit 

I 

length  is  -^.     The  modulus,  E,  is  the  stress  divided 
±j 

W         I 
by  the  strain,  or  —=r  -.  —  ^-,  or 

(6*)     E  =  ^L 

The  measurements  are  to  be  taken  in  kilograms  and 
millimetres. 

Another  method  of  finding  the  modulus  is  by  hang- 
ing weights  from  the  middle  of  a  rectangular  bar  which 
is  supported  near  its  ends  upon  knife  edges,  as  in  Fig. 
239.  If  the  vertical  thickness  of  the  bar  be  A,  its 
breadth,  ft,  and  its  length,  L,  between  the  triangular 
supports,  and  if  a  weight,  W,  lowers  the  middle  of  the 
bar  a  distance  I  it  may  be  shown  by  higher  mathematics 
that  the  modulus,  E,  is 


The  measurements  are  to  be  taken  in  kilograms  and 
millimetres. 

EXERCISE  179.  —  To  find  the  modulus  of  elasticity  of  a  wire 
by  stretching. 

Apparatus.  —  As  shown  in  Fig.  238.  The  wire  is  supported 
by  looping  it  over  a  strong  hook  in  the  ceiling.  The  ends  are 


324  FORCE. 

twisted   together  at  the  bottom  so  as  to  support  the  pan  for 
weights.     To  measure  the  elongation  a  lever  is  used  consisting 
of  a  metre  stick  having  a  nail  through  it,  around  which  the 
wires  are  wound  two   or 
three  times,  the  two  wires 
being   wound    in    opposite 
directions.     The  nail  is 
placed  so  that  its  centre  is 
just  on  the  1  cm.  division. 
Through  the  centre  of  the 
6  cm.  division  a  larger  nail 
passes  and  rests  in  two  sup- 
ports which  are  attached  to  FlG'  238' 
a  block  clamped  to  the  table.     A  darning  needle   projects  6 
cm.  from  the  long  end  of  the  lever,  making  the  long  arm  just 
twenty  times  as  long  as  the  short  one. 

Directions.  —  Measure  the  height  of  the  pointer 
above  the  table  when  enough  weight  is  in  the  pan  to 
make  the  wire  straight.  Add  one  kilogram  and  meas- 
ure the  height  again.  Add  a  second  and  third  weight 
and  measure  the  height  of  the  pointer  each  time.  Re- 
move the  weights  and  see  if  the  zero  reading  remains 
unchanged.  Subtract  each  reading  from  the  next 
larger  and  see  how  they  agree.  If  the  first  is  much 
larger  than  the  rest  it  should  be  discarded,  as  the  wire 
was  not  yet  straight.  Repeat  the  operation  twice  and 
average  all  your  differences.  Divide  the  average  by 
the  ratio  of  the  arms  of  the  lever,  and  you  have  Z,  the 
elongation  in  millimetres  for  a  force  of  one  kilogram. 
Measure  the  length  of  the  wire  from  the  hook  to  the 
point  of  attachment.  Measure  the  diameter  of  the  wire 


YOUNGS  MODULUS  OF  ELASTICITY.  325 

and  multiply  the  square  of  the  diameter  by  .7854  to  get 
the  area  of  one  wire,  which  must,  in  this  case,  be  multi- 
plied by  2  to  get  a.  Compute  the  modulus  by  the 
formula 


EXERCISE  180.  —  To  find  the  modulus  of  elasticity  of  a  bar. 

Apparatus.  —  A  long  rectangular  bar  of  uniform  dimensions, 
resting  on  two  parallel  knife  edges  ;  spherometer  and  sup- 
port, S  (see  Fig.  239). 


FIG.  239. 

Directions.  —  Place  the  spherometer  in  position,  with 
its  legs  passing  through  the  small  holes  in  support,  S, 
winch  is  screwed  fast  to  the  table.  Take  a  zero  reading 
before  adding  any  weights.  Add  a  kilogram  weight 
and  again  take  a  reading  with  the  spherometer.  Add 
successively  three  weights  and  take  a  reading  each 
time.  Remove  the  weights  one  at  a  time  and  take  an- 
other set  of  readings.  The  second  set  should  agree 
fairly  well  with  the  first  if  the  limit  of  elasticity  has 
not  been  exceeded.  Call  the  total  weight  added  W,  and 
the  amount  tne  bar  is  depressed  I.  Measure  the  verti- 
cal thickness,  h,  of  the  bar,  and  its  breadth,  b.  Meas- 
ure the  length  of  the  bar,  L,  between  the  knife  edges. 


326  FORCE. 

Substitute  the  values  obtained  in  the  formula 


SUBSTANCE 

Brass 

Copper 

Iron 

Wood 

Maple 

Oak 

Pine 

EXERCISE  181. 


BREAKING  STRESS 


6.     ELASTICITY. 
MODULUS 

kg. 
sq.  mm. 

9930 
1244!) 
19000 


1021  2.7 

921  5.7 

1113  4.2 

To  measure  the  breaking  stress  of  a  wire. 


£  to 


Apparatus.  —  Pail,  sand,  large  balance,  hammer,  clamp,  fine 
wire. 

Directions.  —  Clamp  a  hammer  to  the  table  so  that 
the  handle  will  project  as  shown 
in  Fig.  240.  Wind  one  end  of 
a  fine  wire  several  times  about 
the  hammer  handle  and  then 
twist  it  securely  fast,  to  the 
clamp.  Fasten  the  other  end 
of  the  wire  to  the  handle  of  the 
pail.  The  wire  should  be  of 
such  length  that  the  pail  will 
hang  about  an  inch  above  the 

FIG.  240.  floor> 

Pour  sand  into  the  pail  a  cupful  at  a  time  until  the 
wire  breaks.  Weigh  the  sand  and  pail  and  take  out  a 


COEFFICIENT  OF  FRICTION.  327 

single  cupful  and  repeat,  pouring  the  sand  in  slowly  so 
as  not  to  put  in  more  than  enough  to  break  the  wire. 
If  it  breaks  where  it  is  fastened  at  either  end  the  ob- 
servation must  be  rejected,  since  any  bend  in  the  wire 
weakens  it. 

Measure  the  diameter  of  the  wire  with  the  microm- 
eter gauge  and  compute  its  cross  section.  The  area 
of  a  circle  is  3.1416  r2.  Calling  the  weight  in  kilo- 
grams W  and  the  area  in  square  millimetres  of  a  cross 
section  a,  the  breaking  stress,  $,  is 


FRICTION. 

178.  Coefficient  of  Friction.  —  The  coefficient  of 
friction  is  the  ratio  of  the  force  required  to  overcome 
friction  to  the  pressure  on  the  surfaces. 

EXERCISE  182.  —  To  find  the  coefficient  of  sliding  friction. 
Apparatus.  —  Planed  board  and  planed   rectangular  block, 
spring  balance. 

Directions.  —  (a)  Lay  the  board  upon  a  level  table, 
fasten  a  spring  balance  to  the  middle  of  one  end  of  the 
block  by  means  of  a  string  and  screw  eye,  and  lay  the 
block  on  the  board. 

Measure  the  force  required  to  keep  the  block  sliding 
along  the  board.  Weigh  the  block.  Put  a  weight  on 
the  block  and  repeat.  Care  must  be  taken  to  keep 
the  string  level. 

(6)  Support  one  end  of  the  board  at  such  a  height 
(to  be  found  by  trial)  that  the  block,  when  once 


328 


FORCE. 


started,  will  continue  to  slide  down  the  inclined  plane. 
If  the  height  of  the  plane  is  AC  and  its  base  CB,  the 

AC 

force  required  to  overcome  friction,  F,  is  —  —  times  the 

C-O 

weight    producing  pressure,  IF,  of  the   block  (see  Fig. 
241). 

It    follows    that 
F      AC 

w=VB  and  the 

coefficient  of   fric- 
tion, /,  is 

(55)    /=*L 
FIG.  241.  W 

EXERCISE  183.  —  To  determine  the  efficiency  of  a  pulley. 
Apparatus.  —  Large  pulley  and  weights,  spring  balance. 

Directions.  —  Weigh  the  weight,  IF,  with  the  balance, 
hang  the  weight  to  one  end  of  the  rope  which  passes  over 
the  pulley  and  attach  the  balance  to  the  other  end. 
Measure  the  force  required  to  keep  the  weight  rising 
after  it  starts.  Call  it  W.  Then  the  work  done 
against  friction  is  W  -  -  W  =  F  and  the  coefficient  of 
friction  is 

(55)    f=*L 
The  efficiency  of  the  pulley  is 


if  the  pulley  is  single,  since  the  distance  through  which 
both  forces  act  is  the  same. 


CHAPTER   XIII. 

HEAT. 
TEMPERATURE. 

EXERCISE  184.  —  To  test  the  zero  point  of  a  thermometer. 
Apparatus.  —  Thermometer,  funnel,  and  jar. 

Directions.  —  The  zero  point  of  a  Centigrade  thermom- 
eter is  the  temperature  of  melting  ice.  Set  the  funnel 
over  the  jar  and  fill  it  with  bits  of  broken  ice.  When 
the  ice  has  been  long  enough  in  the  room  so  that  water 
is  dripping  from  the  funnel,  bury  the  thermometer  in  the 
ice  as  far  as  the  zero,  and  leave  it  till  it  has  gone  as  low 
as  it  will  (say  three  minutes)  and  note  the  reading.  If 
it  should  read  —  .3°  the  error  is  —  .3°  and  the  correc- 
tion is  .3°;  that  is,  .3°  must  be  added  to  the  reading 
of  this  thermometer  at  or  near  0  to  give  the  true  tem- 
perature. 

EXERCISE  185.  — To  test  the  100°  point  of  a  thermometer. 

Apparatus.  —  Thermometer,  flask,  Bunsen  burner,  jacket, 
sheet  of  asbestos,  support.  The  jacket  is  a  large  glass  tube 
passing  through  a  thin  cork,  and  fitting  loosely  in  the  flask. 

Directions.  —  The  100°  point  of  a  Centigrade  ther- 
mometer is  the  boiling  temperature  of  pure  water. 
Water  which  contains  impurities  will  have  a  boiling 
point  slightly  higher,  but  as  the  steam  will  be  pure 
water  it  is  customary  to  take  the  temperature  of  the 
steam.  Fill  a  flask  half  full  of  water,  hang  the  ther- 

329 


330 


HEAT. 


mometer,  protected  from  the  air  by  the  jacket,  from  a 
support  so  that  it  will  hang  about  1  cm.  above  the  liquid 
(see  Fig.  242).  Apply  heat  till  the  water  boils.  After 

it  has  boiled  two  or  three 
minutes  note  the  reading  of 
the  thermometer*  and  read 
the  barometer.  The  ther- 
mometer should  not  project 
far  above  the  top  of  the 
jacket.  The  cork  should  fit 
loosely  upon  the  neck  of  the 
flask,  so  as  not  to  confine  the 
steam. 

The  boiling  point  is  lower 
than  100°  when  the  pressure 
of  the  air  is  less  than  76  cm. 
Hence  a  correction  must  be 
made  for  the  reading  of  the 
barometer.  The  boiling  point 
is  lowered  0.0375°  C.  for  each  millimetre  decrease  in  the 
pressure  of  the  air.  The  true  boiling  point  for  a  pres- 
sure of  b  millimetres  is  therefore 

(57)     t'  =  100°  —  0.0375  (760  —  6) 

Compute  tr  and  compare  it  with  your  observed  value. 
The  difference  is  the  error  of  your  thermometer  near 
100°. 

*  Should  the  thermometer  be  several  degrees  too  low  examine  It  carefully 
to  see  if  some  of  the  mercury  is  not  in  the  little  enlargement  at  the  top.  If  so, 
it  mny  be  brought  back  to  place  by  taking  the  thermometer  in  the  hand  and 
swinging  it  downward  at  arm's  length. 


FIG.  242. 


TEMPERATURE.  331 

EXERCISE  186.  —  To  find  the  temperature  of  your  blood. 
Apparatus.  —  Thermometer. 

Directions.  —  Let  your  assistant  hold  the  thermometer 
in  his  mouth,  keeping  the  lips  closed,  for  about  two  or 
three  minutes.  Read  the  thermometer  to  tenths  of  a 
degree.  Let  your  assistant  take  your  temperature  in 
the  sanle  way.  Compare  the  results  obtained.  Com- 
pare the  amount  of  difference  in  the  results  with  the 
difference  obtained  in  the  pulse  rate  and  in  the 
respiration  found  in  Exercises  170  and  171. 

EXERCISE  187. —  To  find  the  melting  point  of  a  solid 
like  paraffine. 

Apparatus.  —  Thermometer,  glass  tube,  beaker, 
Btinsen  burner. 

Directions.  —  Draw  the  glass  tube  out  at  one  end 
to  a  capillary  tube  with  thin  walls,  melt  some  of 
the  solid  in  a  tin  cup  and  draw  a  portion  of  it 
into  the  tube.  Close  the  tip  of  the  tube  in  the 
flame.  Make  two  little  rubber  bands  by  cutting 
bits  from  the  end  of  a  rubber  tube,  and  by  means 
of  the  bands  attach  the  tube  to  the  thermometer, 

as  shown  in  Fig.  243.    Heat  some  water  in  a  beaker 

FIG. 

while  you  stir  it  with  the  thermometer,  and  watch  243. 
carefully  for  the  instant  that  the  solid  begins  to  melt, 
as  shown  by  its  becoming  transparent.  Read  the  ther- 
mometer, remove  the  flame,  and  watch  for  the  liquid  to 
solidify.  The  thermometer  may  heat  and  cool  a  little 
more  slowly  than  the  substance,  hence  the  average  of 
the  two  readings  should  be  taken  as  the  melting  point 


332 


HEAT. 


of  the  solid.     Repeat  four  times,  always  observing  the 
change  in  the  smallest  part  of  the  tube. 

EXPANSION. 

179.  Coefficient  of  Expansion.  — 'The  ratio  of  the 
increase  in  length  per  unit  length  of  a  substance  for 
one  degree  increase  of  temperature  is  called  the  coeffi- 
cient of  linear  expansion. 

EXERCISE  188.  —  To  find  the  coefficient  of  linear  expansion  of 

a  metal. 

Apparatus. — Box,  spherometer,  cans  of  thin  metal  about 

30  cm.  high  and  8  cm.  in  diameter,  thermometer,  piece  of  plate 
glass.  The  bqx  has  holes  in  one  end  to  fit  the 
legs  of  th^e  spherometer,  and  a  large  hole  for 
th0-  screw" (see  Fig.  244).  Two  blocks  at  right 
angles  define  the  position  of  the  can  in  the  box. 
The  can  should  be  wrapped  in  several  layers  of 
flannel  cloth  to  prevent  loss  of  heat  by  radia- 
tion. It  may  be  sewed  fast  and  left  on  the  can, 
but  must  not  be  allowed  to  get  wet. 

Directions.  —  Set  the  spherometer  in 
place,  measure  the  length,  L,  of  the  can, 
and  fill  the  can  to  within  about  5  mm.  of 
the  top  with  cold  water.  Take  the  tem- 
perature of  the  water,  set  the  can  in  place, 
seeing  that  the  seam  is  exactly  over  a  mark 

of  the  box  and  that  the  can  is  back 
Put  the  plate  glass  on  top  of  the 


i3® 

FIG.  244. 

on  the  bottom 

against  the  blocks. 

can,  bring  the  spherometer  screw  down  upon  it,  and  read 

the  spherometer.     Raise  the  screw,  remove  the  glass  and 

can,  take  the  temperature  again,  empty  the  can  and  fill 


SPECIFIC  HEAT.  333 

it  with  boiling  water  from  a  teakettle  or  pail,  take 
its  temperature  and  replace  it  in  its  former  position. 
Again  read  the  spherometer,  remove  the  can,  and  take 
the  temperature.  The  average  of  the  temperature  just 
before  putting  the  can  in  place  and  just  after  removing 
it  may  be  taken  as  the  temperature  of  the  metal  at  the 
time  the  spherometer  reading  was  made.  The  difference 
in  the  two  spherometer  readings  is  the  expansion,  Z,  of 
the  can  and  the  difference  between  the  temperatures  of 
the  hot  and  cold  water  is  the  change,  t,  in  temperature 
of  the  metal.  The  coefficient  of  linear  expansion,  e,  is : 

(58)     e  =  — 
Lt 

We  assume  in  this  experiment  that  the  temperature 
of  the  metal  is  the  same  as  that  of  the  water.  If  the 
metal  is  thin  and  well  protected  by  flannel  this  will  be 
very  nearly  the  case. 

Care  must  be  taken  not  to  spill  the  water  in  the  box, 
as  the  wood  will  expand  when  wet  and  spoil  the  results. 

7.  COEFFICIENTS  OF  EXPANSION. 

SUBSTANCE  COEFFICIENT    SUBSTANCE      COEFFICIENT 
Aluminum  0.000023         Glass  0.000009 

Brass  0.000018         Iron  0.000012 

Copper  0.000017         Platinum  0.000009 

SPECIFIC   HEAT. 

180.  Heat  Capacity.  —  Bodies  differ  in  their  capac- 
ity for  heat,  that  is,  in  the  amount  of  heat  required  to 
raise  their  temperature  one  degree. 


334  HEAT. 

The  capacity  of  one  gram  of  a  substance  divided  by 
the  capacity  of  one  gram  of  water  (one  calorie)  is  called 
the  specific  heat  of  the  substance.  Water  has  the  high- 
est specific  heat  of  any  known  substance.  Hence,  if  we 
take  the  specific  heat  of  water  as  unity  the  specific  heats 
of  other  substances  are  expressed  by  fractions  differing 
considerably  in  value  for  different  substances. 

The  principle  of  the  method  of  mixtures  is  as  follows : 
If  two  substances  at  different  temperatures  be  mixed 
together  the  cool  body  will  take  heat  from  the  warmer 
till  they  are  of  the  same  temperature.  The  quantity  of 
heat  lost  by  the  hot  body  must  equal  that  given  the  cold 
body.  Account  must  also  be  taken  of  the  heat  radiated 
to  the  surrounding  air  and  of  that  conducted  to  the 
vessel  in  which  the  two  substances  are  mixed  and  to  the 
thermometer. 

EXERCISE  189.  —  To  determine  the  specific  heat  of  a  metal  by 
the  method  of  mixtures. 

Apparatus. — Calorimeter,  thermometer,  shot  or  bits  of 
wire,  test  tube,  beaker. 

Directions.  —  Weigh  out  M  grams  of  the  substances 
whose  specific  heat  is  sought.  Weigh  in  the  calorim- 
eter, which  may  be  a  copper  beaker  or  a  tin  cup,  a 
mass,  m,  of  water.  If  the  temperature  of  the  water  is 
a  little  less  than  that  of  the  room,  about  as  much  heat 
will  be  gained  from  the  air  of  the  room  as  is  lost  during 
the  operation. 

Put  the  shot  or  bits  of  metal  in  a  test  tube  and  set 
the  tube  in  a  beaker  of  boiling  water  till  the  metal  has 


SPECIFIC  HEAT.  335 

had  time  to  take  the  temperature  of  the  water.  Read 
the  barometer  and  compute  the  temperature.  Have 
ready  the  cup  of  cool  water  with  a  thermometer  in  it. 
Note  the  temperature  of  the  water.  Pour  the  metal 
from  the  test  tube  into  the  water,  being  careful  not  to 
let  any  water  from  the  outside  of  the  test  tube  fall  into 
the  cup.  Now  stir  the  shot  and  water  with  the  ther- 
mometer and  note  the  highest  temperature  which  the 
mixture  reaches  before  it  begins  to  cool  by  radiation. 
Call  the  temperature  of  the  shot  T,  that  of  the  water  £, 
and  that  of  the  mixture  t'.  Call  the  specific  heat  of 
water  s.  It  is  by  definition  equal  to  1. 

In  any  transfer  of  heat  the  amount  of  heat  transferred 
is  equal  to  the  mass  times  the  specific  heat  times  the 
change  in  temperature  of  the  substance.  In  this  case 
the  wrater  has  received  ms  (t1  —  f)  calories  and  the  cal- 
orimeter and  thermometer  have  received  an  amount 
which  we  call  e  (t1  —  f),  where  e  is  the  amount  of  water 
which  is  equivalent  in  heat  capacity  to  the  calorimeter 
and  thermometer.  The  metal  has  lost  MS  (T—  t1)  cal- 
ories. But  the  two  quantities  of  heat  are  equal,  hence : 

MS  (T—  t')  =  (ms  +  e)  (t'—  t) 
(59)     S  =  (ms  +  e)  (f  —  t)  /  M (T  —  V} 

To  find  e,  let  the  calorimeter  and  thermometer  take 
the  temperature,  tv  of  the  room.  Pour  into  the  calorim- 
eter m1  grams  of  water  at  a  temperature  2\  (say  30°), 
stir  with  the  thermometer,  and  note  the  temperature 
attained,  £.  Then 


336  HEAT, 


(60}     e  =  w1(ri- 

Substitute  your  values  in  the  formula  (59)  and  com- 
pare the  result  with  the  value  given  in  Table  8. 

8.     SPECIFIC  HEATS. 

SUBSTANCE     SPECIFIC  HEAT      SUBSTANCE  SPECIFIC  HEAT 
Aluminum               0.212                   Iron  0.113 

Brass  0.0891  Lead  0.032 

Copper  0.0931  Mercury  0.033 

Glass  0.02  Paraffine  0.589 

HYGKOMETRY. 

181.  Hygrometry  has  to  do  with  the  amount  of 
water  present  at  any  time  in  the  atmosphere.  By  the 
absolute  humidity  is  meant  the  number  of  grams  of 
water  present  in  one  cubic  metre  of  air.  By  the  relar 
tive  humidity  is  meant  the  ratio  of  the  amount  of  vapor 
in  the  air  to  the  amount  required  to  saturate  the  air  at 
the  given  temperature.  The  deiv  point  is  the  tempera- 
ture at  which  the  vapor  now  in  the  air  would  saturate 
the  air  and  be  deposited  as  dew  or  frost,  according  to 
whether  the  dew  point  falls  above  or  below  zero. 

EXERCISE  190.  —  To  find  the  humidity  and  dew  point  with 
wet  and  dry  thermometers. 

Apparatus.  —  Wet  and  dry  bulb  thermometers.  Two  ther- 
mometers of  about  the  same  size  are  hung  a  few  centimetres 
apart.  To  one  of  them  is  attached  a  wick  of  cotton  cloth  which 
dips  in  a  dish  of  water. 

The  evaporation  of  the  water  cools  the  bulb  of  the  thermom- 
eter, making  it  read  lower  than  the  dry  thermometer.  The 


HYGEOMETRT.  337 

more  water  there  is  in  the  air,  the  less  evaporation  there  will 
be.  When  the  air  is  saturated  (relative  humidity  =  1)  the  two 
thermometers  will  read  alike. 

Directions.  —  See  that  the  wick  is  moist.  Read  both 
thermometers,  record,  wait  three  minutes,  read  again, 
and  take  a  third  reading  three  minutes  after  the  second. 
Currents  of  air  may  affect  the  readings,  but  the  differ- 
ence should  be  nearly  constant.  Call  the  reading  of 
the  dry  thermometer  £,  of  the  wet  one  t'.  (a)  Look  up 
in  Table  9  the  absolute  humidity,  A',  for  the  tempera- 
ture t'.  It  has  been  found  that  the  absolute  humidity, 
H,  of  the  air,  which  is  at  temperature  £,  is,  in  a  closed 
room, 

(61)    H=h'  —  0.96  (t  —  t') 

(6)  If  h  be  the  humidity  of  saturated  air  for  temper- 
ature £,  as  given  in  Table  9,  then  the  relative  humidity 
is 

(62)    HI  =  H/  h 

(e)  The  dew  point,  T,  for  any  absolute  humidity,  If, 
may  be  found  from  Table  9. 

Compare  your  results  (<?)  and  (a)  with  the  results 
obtained  at  the  same  hour  by  some  other  student  who 
has  used  the  method  described  in  the  following  exercise. 

EXERCISE  191.  —  To  find  the  dew  point  and  absolute  humid- 
ity of  the  air  by  cooling  the  air. 

Apparatus.  —  Beaker,  water,  ice  water,  thermometer. 

Directions.  —  Fill  the  beaker  one  third  full  of  water 
at  the  temperature  of  the  room.  Add  ice  water,  a  little 


338  HEAT. 

at  a  time,  stirring  with  the  thermometer.  Watch  care- 
fully for  the  first  deposit  of  dew  on  the  outside  of  the 
beaker,  read  the  thermometer,  and  watch  for  the  dew 
to  disappear  and  again  read  the  thermometer.  The 
average  reading,  T,  is  the  dew  point.  The  absolute 
humidity,  H,  corresponding  to  the  dew  point,  T,  may 
be  found  in  Table  9. 

9.    HUMIDITY. 


t 

A 

t 

h 

t 

h 

t 

h 

10° 

9.4 

15° 

12.8 

20° 

17.2 

25° 

22.9 

11 

10.0 

16 

13.6 

21 

18.2 

26 

24.2 

12 

10.6 

17 

14.5 

22 

19.3 

27 

25.6 

13 

11.3 

18 

15.1 

23 

20.4 

28 

27.0 

14 

12.0 

19 

16.2 

24 

21.4 

29 

28.6 

CHAPTER  XIV. 
ELECTRICITY. 

EXERCISE  192.  —  To  measure  the  resistance  of  a  conductor  by 
substitution. 

Apparatus. — Galvanometer,  resistance  box,  switch,  gravity 
battery. 

The  resistance  box  (Fig.  245)  is  a  wooden  case  containing  a 
set  of   coils   of   German-silver  wire,  _R,  J?', 
etc.,  of   known   resistance,  which   are   con- 
nected to  blocks  of  brass,  B. 

The  blocks  may  be  connected  by  means  of 
the  brass  plugs  P1?  P2,  etc.  The  blocks  and 
plugs  are  so  large  that  their  resistance  may  be 
neglected  in  comparison  with  the  resistance 
of  the  coils.  When  all  the  plugs  are  in-  FIG.  245. 

serted  the  resistance  of  the  box  is  reckoned  as  0.  When  P1  is 
removed  the  resistance  is  R.  The  resistance  of  the  several 
coils  is  marked  upon  the  top  of  the  box. 

Directions.  —  Connect 
the  battery,  B,  galvanom. 
eter,  6r,  resistance  box,  Rv 
unknown  resistance,  J?2, 
switch,  >Sr,  and,  if  needed, 
additional  resistance,  R,  as 
shown  in  Fig.  246.  When  the  arm  of  the  switch,  8,  is 
on  O  the  current  flows  through  Rr  When  it  is  on  A  it 
flows  through  R2.  The  galvanometer  should  be  placed 

339 


340 


ELECTEICITY. 


with  its  coils  north  and  south,  the  needle  also  north  and 
south  and  reading  zero.  Sometimes  readings  are  taken 
from  a  pointer  at  right  angles  to  the  needle.  Send  the 
current  through  ~R2  and  read  the  deflection  on  the  gal- 
vanometer. If  it  is  more  than  60°,  additional  resistance, 
R,  should  be  put  in  the  circuit.  Read  both  ends  of 
the  needle  and  take  the  average.  Having  determined 
the  deflection  produced  when  R^  is  in  circuit,  throw  Ml 
in  circuit  and  vary  it  by  changing  the  resistance  in  the 
box  till  the  deflection  is  exactly  the  same  as  before. 
It  is  then  obvious  that  M2  =  Rr 

EXERCISE  193.  —  To  measure  the  resistance  of  a  conductor 
with  the  Wheatstone's  bridge. 

Apparatus.  —  Galvanometer,  resistance  box,  Wheatstone's 
bridge,  gravity  battery. 

The  principle  of  the  bridge  is  illustrated  in  the  typical  dia- 

gram, Fig.  247.  A  cur- 
rent from  the  battery 
divides  at  A,  part  going 


to  B  throuh 


J? 


another  part  through  jR3, 
J?4.  It  can  be  proved 
that  if  Rl:E2  =  R3:  R4 
no  current  will  flow 
through  the  galvanom- 
eter between  C  and  _D. 
The  converse  is  also  true 
that  if,  when  a  current  is  flowing  from  A  to  B,  no  current  flows 
through  the  galvanometer  the  four  resistances  are  in  the  ratio 
named.  If,  then,  we  know  Hl  and  the  ratio  of  _B4  to  J23  we  can 
at  once  compute  _R2-  It  *s 

(63)  BZ  =  B1XEJHZ 


FIG. 


RESISTANCE.  341 

The  bridge  shown  in  Fig.  248  is  essential  in  principle  with 
the  ideal  bridge,  Fig.  247.  For  convenience  in  adjusting  the 
ratio  of  J24  :  Ez  these  re- 
sistances are  composed 
of  a  platinum  or  Ger- 
man-silver wire  of  uni- 
form diameter,  one  me- 
tre in  length,  stretched 
between  the  heavy  cop- 
per washers,  A  and  B. 
A  metre  stick  fastened  FlG 

beside  the  wire  enables 

us  to  read  at  once  the  ratio  of  the  parts  into  which  the  wire  is 
divided  by  the  contact  _D. 

Directions.  —  Connect  the  battery  to  points  A,  B,  and 
the  galvanometer  to  points  (7,  D.  The  coils  of  the 
galvanometer  should  stand  north  and  south  and  the 
needle  should  read  zero.  The  key,  K,  may  .be  left 
closed  when  a  gravity  cell  is  in  circuit.  With  any 
other  battery  the  circuit  should  never  be  kept  closed 
except  when  the  readings  are  being  taken.  Connect 
the  resistance  box  as  ^  and  the  unknown  resistance  as 
jR2.  See  that  the  contacts  are  clean  and  firm.  Old 
wires  should  be  scraped  with  a  knife.  If  the  wires  are 
not  smaller  than  No.  16  their  resistance  may  be  neg- 
lected. Put  some  resistance  in  the  box,  close  K,  and 
make  contact  with  D  at  the  middle  of  the  wire. 
Notice  which  way  the  galvanometer  deflects  and  move 
the  contact  first  one  way  and  then  the  other  till  you 
find  a  position  such  that  the  galvanometer  deflects  the 
other  way.  If  you  have  moved  far  to  the  left  of  50 


342  ELECTRICITY. 

the  resistance  Rl  is  too '  small ;  if  to  the  right,  it  is  too 
large.  Compute  roughly  what  7?2  is  and  put  about  that 
amount  in  Rr  The  contact  will  then  fall  near  the 
middle.  Near  50  an  error  in  setting  the  contact  of 
1  cm.  will  be  an  error  of  about  2  in  50,  or  4  per  cent, 
while  a  like  error  at  90  will  be  an  error  of  2  in  10,  or 
20  per  cent.  Find  the  position  for  D  where  the  galva- 
nometer seems  to  come  to  zero,  move  it  a  little  to  the 
right  till  the  needle  deflects,  then  move  it  a  like 
amount  to  the  left  and  see  if  it  deflects  about  the  same 
amount  in  the  opposite  direction.  When  satisfied  that 
you  have  the  position  for  zero  deflection  read  the  posi- 
tion of  D  on  the  metre  stick.  If  it  reads  from  left  to 
right  the  reading  may  be  considered  7?3,  while  100 
minus  the  reading  is  J?4,  since  we  are  concerned  only 
with  the  ratio  of  R±  to  Ry  not  with  their  absolute 
values. 

If  Rl  was  less  than  R2  take  another  observation, 
making '  El  greater  than  E^.  If  possible  make  an 
observation  with  Rl  nearly  equal  to  Ry 

182.  Specific  Resistance.  —  Two  wires  of  the  same 
length  and  diameter  will  have  the  same  resistance  if 
they  are  of  the  same  substance,  different  resistances  if 
of  different  substances.  The  resistance  of  unit  length 
and  unit  cross  section  of  a  wire  of  any  substance  is  the 
specific  resistance  of  that  substance.  The  specific 
resistance,  r,  of  a  wire  having  length  .L,  cross  section 
(?,  and  resistance  R  is 


RESISTANCE.  343 


(64)     r  = 

±J 

EXERCISE  194.  —  To  find  the  specific  resistance  of  a  metal. 
Apparatus.  —  Wheatstone's  bridge,   resistance   box,   galva- 
nometer, fine  wire,  micrometer,  metre  stick. 

Directions.  —  Measure  the  resistance,  R,  length,  L, 
and  diameter  of  the  wire.  From  its  diameter  compute 
its  cross  section,  a.  Compute  the  specific  resistance 
from  the  formula 


183.  Temperature  Coefficient  of  Resistance.  —  The 

resistance  of  a  metal  increases  with  its  temperature.  In 
the  case  of  some  alloys,  like  German  silver,  the  change 
is  slight,  making  them  suitable  for  standards  of  resist- 
ance. In  the  case  of  carbon  the  resistance  decreases  as 
the  temperature  rises. 

The  temperature  coefficient  of  resistance  is  the  rate 
of  increase  in  resistance  per  ohm  of  resistance  for  one 
degree  rise  in  temperature.  If  a  conductor  has  a  resists 
ance  R  at  temperature  £,  and  R'  at  temperature  tr,  its 
coefficient  between  those  two  temperatures  is 

(65)    c=    *'-* 
R  (f  —  0 

EXERCISE  195.  —  To  find  the  temperature  coefficient  of  resist- 
ance of  a  metal. 

Apparatus.  —  Coil  of  fine  wire,  beaker,  hot  water,  bridge, 
galvanometer  and  resistance  coils,  thermometer. 

Directions.  —  Support  the  coil,  with  the  thermometer 


344 


ELECTRICITY. 


in  it,  in  the  beaker.  Measure  the  resistance,  R,  of  the 
coil  and  note  the  temperature,  t.  Pour  enough  hot 
water  into  the  beaker  to  cover  the  coil.  Note  the  tem- 
perature, t',  and  measure  the  resistance,  R'.  The  point 
of  greatest  difficulty  will  be  to  determine  the  exact  tem- 
perature at  the  instant  the  reading  is  taken.  If  the  coil 
be  lifted  out  and  the  water  stirred  and  its  temperature 
taken  at  the  instant  the  bridge  is  adjusted,  the  temper- 
ature should  not  be  far  from  right. 


10.  KESISTANCES. 


SPECIFIC 

TEMPERATURE 

RESISTANCE 

COEFFICIENT 

0.0000029 

0.38 

0.0000016 

0.39 

0.0000209 

0.04 

0.0000097 

0.53 

SUBSTANCE 

Aluminum 
Copper 
German  silver 
Iron 


EXERCISE  196.  —  To  measure  the  resistance  of  a  battery  with 
the  Wheatstone's  bridge. 

Apparatus.  —  Bridge,  small  induction  coil,  telephone. 

Directions.  —  If  we  should  connect  a  battery  as  R^  in 

the  bridge,  the  current 
from  the  battery  itself 
would  flow  through 
the  galvanometer 
even  if  the  bridge 
were  balanced.  To 
avoid  this  difficulty 
substitute  for  the 
battery  in  the  bridge  a  small  induction  coil,  I  (Fig. 


FIG.  249. 


POTENTIAL  DIFFERENCE.  345 

249),  giving  an  alternating  current  and  for  the  galva- 
nometer a  telephone,  T.  The  telephone  will  be  affected 
by  the  alternating  current  from  the  induction  coil,  but 
not  by  the  direct  current  from  the  battery  at  R2.  The 
induction  coil  should  be  a  small  one  of  low  resistance. 
It  may  be  run  by  a  storage  battery  or,  for  a  little  while, 
by  two  Leclanche  batteries. 

Adjust  the  contact  for  silence  in  the  telephone.  In 
all  other  respects  the  directions  given  for  Exercise  193 
apply  for  this. 

EXERCISE  197.  —  To  measure  the  potential  difference  of  a 
battery. 

Apparatus.  —  Battery,  ammeter,  resistance  box.  The  re- 
sistance of  the  ammeter,  -B',  and  battery,  JK",  must  have  been 
determined  previously. 

Directions. — Connect  the  battery  in  series  with  the 
ammeter  and  resistance  box.  With  a  resistance,  R,  in 
the  box  read  from  the  ammeter  the  current,  i,  in  am- 
peres. By  Qhm's  law 


CHAPTER   XV. 
SOUND. 

184.  Velocity  of  Sound.  —  Sound  emitted  by  a  body 
making  n  vibrations  per  second  travels  nl  metres  per 
second  in  a  medium  where  the  length  of  a  wave  is  I 
metres. 

EXERCISE  198.  — To  find  the  velocity  of  sound  in  air. 

Apparatus.  —  Tuning  fork  making  a  known  number  of  vi- 
brations, long  glass  tube  fitted  with  an  outlet  tube  and  pinch 
cock,  thermometer. 

Directions.  —  Fill  the  tube  nearly  full  of  water,  strike 

the  fork  against  the  end  of  a  heavy  block  of  wood  and 

hold  it  over  the  mouth  of   the   tube.     Let   out 

•tfS* 

a  little  of  the  water  and  again  try  the  fork. 
When  the  tube  sounds  with  a  considerable  volume 
of  tone,  measure  the  length  for  each  trial.  When 
the  volume  of  sound  begins  to  diminish  try  to 
judge  which  trial  gave  the  loudest  sound,  fill  the 
JN'  tube  to  a  little  above  the  point  where  the  sound 
seemed  loudest,  and  repeat  till  you  are  sure  you 
have  a  maximum.  Mark  the  spot,  JV^Fig.  250), 
on  the  tube  where  the  sound  was  loudest  with  a 
FIG.  rubber  band  or  a  paper  label. 

At  a  point,  Nf,  a  little  more  than  three  times 
as  far  from  the  top  as  the  first  point,  should  be  found 
a  second  point  where  resonance  occurs. 

346 


VELOCITY  OF  SOUND.  347 

(34}     NN>  =  A  I 
(33)     v  =  nl 

from  which  formula  we  may  easily  compute  the  velocity 
of  sound  for  the  given  temperature,  since  we  know  the 
number  of  vibrations,  n,  of  the  fork. 

Note  the  temperature,  £,  of  the  room.  Compare 
your  result  with  the  value  given  in  Table  11  for  that 
temperature. 

11.  VELOCITY  OF  SOUND 


in  centimetres  per  second. 

SUBSTANCE 

VELOCITY 

SUBSTANCE 

VELOCITY 

Air  0° 

33,220 

Hydrogen,  0° 

126,600 

16° 

34,179 

Water,  4° 

140,000 

18° 

34,267 

Brass 

350,000 

20° 

34,415 

Aluminum 

510,400 

22° 

34,532 

Copper 

356,000 

24° 

34,649 

Glass 

506,000 

28° 

34,881 

Iron 

509,300 

EXERCISE  199. — To  determine  the  velocity  of  sound  in  a 
solid. 

Apparatus.  —  Large  glass  tube,  long  rod  of  wood  or  metal, 
fine  cork  filings,  cloth  and  rosin,  metre  stick.  The  glass  tube 
(see  Fig.  251)  is  fitted  at  one  end  with  a  piston  for  varying  the 
length  of  the  enclosed  air  column.  Cork  filings  are  distrib- 
uted evenly  along  the  length  of  the  tube.  The  rod  is  firmly 
clamped  exactly  at  its  middle  point.  To  one  end  of  the  rod  is 
cemented  a  disk  of  cardboard  or  cork  of  a  size  to  fit  loosely  in 
the  glass  tube.  If  a  hole  is  made  in  the  disk  to  fit  the  rod  it 
will  be  held  more  firmly  to  the  rod. 

Directions.  —  Support  the  tube  so  that  the  end  of  the 
rod  carrying  the  disk  projects  a  few  centimetres  within 


348  SOUND. 

it.  "With  the  cloth,  on  which  has  been  placed  some 
powdered  rosin,  rub  the  rod  lengthwise  so  that  a  sharp 
note  will  be  produced  by  the  longitudinal  vibrations. 
The  disk  will  set  the  air  in  the  tube  in  vibration,  the 
waves  will  be  reflected  from  the  piston,  forming  station- 
ary waves  with  nodes  half  a  wave  length  apart.  The 
cork  filings  will  be  disturbed  at  the  points  of  greatest 
activity  and  remain  at  rest  at  the  nodes.  The  difference 
will  be  made  more  evident  if  the  tube  is  turned  about 
30°,  so  that  the  filings  lie  to  one  side  of  the  bottom  of 
the  tube  ready  to  slip  down  at  the  slightest  disturbance. 
After  the  note  has  been  sounded  they  will  present  the 


FIG.  251. 

appearance  shown  in  Fig.  251.  Adjust  the  piston  till 
the  proper  length  has  been  found  to  produce  an  exact 
number  of  waves  and  consequently  a  sharp  indication 
with  the  cork  dust.  The  antinodes  are,  like  the  nodes, 
half  a  wave  length  apart.  The  rod  has  a  node  at  the 
centre  and  an  antinode  at  each  end ;  it  is  therefore  half 
a  wave  length  long.  Measure  la ,  the  distance  between 
two  antinodes,  by  measuring  the  distance  between  two 
sharply  defined  antinodes  near  the  ends  of  the  tube  and 
dividing  the  distance  between  by  the  number  of  spaces 
included.  Measure  Zs,  the  length  of  the  rod,  and  note 
the  temperature,  £,  of  the  room. 

Look  up  va,  the  velocity  of  sound,  for  temperature,  t, 
in  Table  11. 


PITCH  OF  A    FOEK.  349 

Then  since  the  velocity  in  any  medium  is  propor- 
tional to  the  wave  length  in  that  medium,  the  velocity 
of  sound  in  the  rod,  va,  is 

(38)     vs  =  ^-va 

la 

EXERCISE  200.  —  To  find  the  vibration  number  of  a  tuning 
fork. 

Apparatus.  —  Two  tuning  forks,  the  vibration  number  of  one 
of  which  is  known,  resonance  jar,  bit  of  soft  wax. 

Directions.  —  Pour  water  into  the  jar  till  it  is  in  res- 
onance with  the  standard  fork.  The  fork  to  be  tested 
must  have  nearly  the  same  frequency  as  the  standard. 
The  jar  will  then  be  in  resonance  for  both  forks.  Strike 
both  forks  at  once  and  hold  them  over  the  jar.  If  they 
are  not  in  perfect  unison  beats  will  be  heard.  Count 
the  beats  in  a  second  by  the  clock. 

If  the  standard  fork  has  a  frequency,  n',  and  there  are 
m  beats  per  second,  the  frequency,  n,  of  the  other  fork 
is 

(66)     n  —  n1  ±  m 

To -find  whether  the  plus  or  the  minus  sign  is  to  be 
used  in  this  formula,  reduce  the  time  of  vibration  of 
the  fork  you  are  testing  by  fastening  a  bit  of  wax  to 
one  prong.  If  the  effect  is  to  diminish  the  number  of 
beats  the  fork  is  lower  than  the  standard,  if  to  increase 
it,  higher.  Check  this  result  by  removing  the  wax  from 
that  fork  and  putting  it  on  the  standard  fork. 


CHAPTER   XVI. 
LIGHT. 

PHOTOMETRY. 

185.  The  Intensity  of  Illumination  at  any  point 
varies  directly  with  the  intensity  of  the  source  of  light 
and  inversely  as  the  square  of  the  distance  of  the  point 
from  the  source  of  light.  If  two  surfaces,  sv  «2,  at 
distance  dv  d^  from  two  sources  of  light,  Sv  $2,  are 
equally  illuminated,  sl  receiving  light  only  from  S1  and 
s2  only  from  $2,  then 


If  S1  be  a  standard  candle  burning  7.78  grams  per 

hour  the  candle  power 
of  #2  is 

(67)     S2  =  d^  /  dj 

EXERCISE  201.  —  To 
measure  the  candle  power 
of  a  lamp  with  Rumford's 
FIG.  252.  photometer. 

Apparatus.  —  Paper  screen,  wooden  rod,  paraffine  candle  of 
size  sixes. 

Directions.  —  Place  the  rod  a  few  centimetres  in  front 
of  the  screen  (see  Fig.  252)  and  set  the  candle  about  a 
metre  from  the  screen.  Move  the  lamp  till  the  two 

350 


INDEX  OF  REFBACT10N.  351 

shadows  are  side  by  side  and  of  equal  density.  Shadow 
sl  (if  the  room  be  dark)  is  illuminated  only  by  Sv 
shadow  s2  only  by  S2.  Measure  dv  the  distance  from 
S1  to  sv  and  d^  the  distance  from  Sz  to  «2,  and  substi- 
tute in  the  formula  £2  =  df  /  c?22.  Change  dl  and  find 
a  corresponding  value  for  dy  Find  thus  four  values  for 
$2  and  take  their  average. 

EXERCISE  202.  —  To  measure  the  candle  power  of  a  lamp 
with  Joly's  photometer. 

Apparatus.  —  Candle,   metre   stick,   paraffine   blocks.     Two 
rectangular  blocks  of  par- 
affine,S     .   (Fig.  253),  2 


cm.  high  and  1  cm.  square        »H 


stand  side   by  side   on   a  FlG-  253- 

support,  and  are  separated  by  a  card  which   reaches   to  the 

front  edge  of  the  paraffine  blocks. 

Directions.  —  Set  the  lamp  and  candle  at  opposite 
ends  of  the  table  and  move  the  support  till  the  paraffine 
blocks  Sp  «9,  are  equally  illuminated.  Measure  dv  d2 
and  substitute  in  the  formula 

(67)    S2  =  tV/<V 

Change  the  distance  of  the  lamp  from  the  candle  and 
make  another  determination. 

EEFRACTION. 

186.  Index  of  Refraction.  —  The  index  of  refrac- 
tion of  a  substance  referred  to  air  is  the  ratio  of  the 
velocity  of  light  in  air  to  its  velocity  in  the  given 
substance. 


352  LIGHT. 


(35)    n  = 

V2 

EXERCISE  203.  —  To  find  the  index  of  refraction  of  glass. 

Apparatus.  —  Piece  of  heavy  plate  glass  with  true  edges, 
sheet  of  paper,  board,  three  pins,  compass. 

Directions.  —  Draw  on  the  paper  two  lines  at  right 
angles  to  each  other  through  S  (Fig.  254).  Place  one 
edge  of  the  plate  glass  against  one  of  the  lines.  Stick 
a  pin  vertically  on  the  other  line  against  the  glass  as  at 
&  With  the  eye  near  the  paper  at  E  the  pin  may  be 

seen  on  the  line  through  the 
glass.  Place  a  second  pin 
at  R  and  move  the  eye 
toward  E1  till  the  image  of 
8  appears  in  line  with  R 
and  the  eye  and  set  a  third 
pin  at  Q  exactly  in  line  with 
FIG.  254.  .RandiS".  Remove  the  glass 

and  draw  SR\  the  direction  of  the  light  in  glass,  and 
RQ,  its  direction  in  air.  Produce  QR  to  meet  SO  at 
jS1,  the  image  of  S.  With  a  radius  SO  and  S  as  a 
centre  draw  the  arc  00',  which  is  the  front  of  a  wave 
from  S,  lying  wholly  in  glass.  With  S'  as  a  centre  and 
a  radius  S'R  draw  RR',  which  is  the  front  of  a  wave 
lying  wholly  in  air. 

It  is  evident  that  while  the  wave  was  travelling  from 
0'  to  R  in  glass  it  travelled  from    0  to  R  in  air.     It 

follows  that 

v,       OH' 


INDEX  OF  REFRACTION. 


353 


and  we  have  only  to  measure   OR'  and   O'R  with  the 
dividers  and  scale. 

EXERCISE  204.  —  To  find  the  index  of  refraction  of  water. 
Apparatus.  —  Tin  cup,  metre  stick,  try-square,  clamp,  shal- 
low box,  large  sheet  of  paper,  compasses,  needle  and  block. 

Directions.  —  Place  the  tin  cup  in  one  corner  of  the 
box  (see  Fig.  255),  and  clamp  the  metre  stick  in  a 
vertical  position  at  the  other  end.  Stick  the  needle 
in  the  block,  set  the 
block  over  the  cup 
with  the  needle  pro- 
jecting downward. 
Fill  the  cup  with 
water  till  the  surface 
just  touches  the  nee- 
dle. Remove  the 
block  and  place  the 
eye  near  the  metre 
stick  in  such  a  posi- 
tion that  the  corner, 
at  the  bottom  of  the  cup,  at  S,  is  just  visible  over  the  top 
of  the  cup.  Place  the  try-square  against  the  metre  stick, 
and  when  it  is  in  line  with  the  top  of  the  cup  and  Sf, 
the  image  of  S,  read  the  scale.  Measure  the  needle  to 
find  the  distance  of  the  water  from  the  top  of  the  cup. 
Measure  also  the  depth  of  the  cup  on  the  inside. 

Lay  off  all  of  these  dimensions  on  the  sheet  of  paper 
as  accurately  as  possible  and  draw  the  lines  shown  in 
Fig.  255.  It  is  evident  that 


FIG.  255. 


354  LIGHT. 


When  the  apparatus  is  in  position  the  index  of  refrac- 
tion of  some  other  liquid,  as  alcohol,  may  be  found  by 
making  but  one  measurement,  TE,  constructing  the 
diagram  and  measuring  OR'  and  O'H. 

12.    INDICES  or  KEFRACTION. 

SUBSTANCE  INDEX  SUBSTANCE  INDEX 

Alcohol  1.36  Glass  1.66 

Carbon  bisulphide      1.65  Water  1.34 

LENSES. 

EXERCISE  205.  —  To  find  the  focal  distance  of  a  convex  lens. 
Apparatus.  —  Metre   stick,   paper   or  ground  glass  screen, 
candle,  mirror. 

Directions.  —  (a)  First  method.     Hold  the  lens  be- 
fore an  open  window  and  move  the  screen  toward  or 

from    it    till    a 
clear    image    of 
s~~  if          s1      a   distant  spire, 

FIG.  256.  tree,  or  cloud  is 

obtained  on  the  screen  (see  Fig.  256).  The  distance  / 
from  the  screen  to  the  optical  centre  of  the  lens  is  the 
focal  distance  of  the  lens.  The  optical  centre  of  a 
double-convex  or  double-concave  lens  is  at  the  centre 
of  the  lens  if  both  sides  are  of  equal  curvature.  In 
a  plano-convex  or  plano-concave  lens  the  optical  centre 
is  at  the  centre  of  the  curved  surface. 

(6)  Second  method.     Place  a  candle  or  other  bright 


LENSES. 


355 


FIG.  257. 


object,  S  (Fig.  257),  near  a  screen,  s,  and  at  one  edge 
of  it.  Hold  a  plane  mirror  behind  the  lens  and  move 
lens  and  mirror  together  till  a  clear  image  of  the  candle 
appears  on  the  screen.  The  distance  of  the  image  or 
object  from  the  centre  of 
the  lens  is  the  focal  dis- 
tance of  the  lens,  for  in 
passing  once  through  the 
lens  the  rays  are  made 
parallel,  and  in  passing 
again  through  the  lens  these  parallel  rays  are  brought 
together  at  the  principal  focus.  If  the  screen  were  not 
at  the  principal  focus  the  distances  SO  and  AS"  0  could 
not  be  equal. 

(e)  Third  method.     Place  the  lens  upon  the  table 
between  the  candle  and  the  screen  and  move  candle  and 

screen  toward  or  from 
it  at  the  same  rate  till 
a  sharp  image  of  the 
candle  is  seen  upon  the 
screen  (see  Fig.  258). 
The  candle  and  screen  are  now  at  the  secondary  foci 
and  the  distance  SS1  is  four  times  the  focal  distance  of 
the  lens.  This  method  does  not  require  that  we  should 
know  the  optical  centre  of  a  lens  if  instead  of  making 
SO  and  AS"  0  equal  we  make  the  length  of  object  and 
image  equal.  It  will  thus  serve  for  measuring  the  focal 
distance  of  a  system  of  lenses,  like  compound  photo- 
graphic lenses  whose  optical  centre  is  not  easily  found. 


FIG- 


356  LIGHT. 

MAGNIFICATION. 

EXERCISE  206.  —  To  measure  the  magnifying  power  (a)  of  a 
telescope,  (b)  of  a  microscope. 

Apparatus.  —  Cardboard  scale,  ruled  to  decimetres  and 
centimetres,  fastened  to  the  wall  about  two  metres  from  the 
floor  so  as  to  be  above  the  heads  of  passers-by,  millimetre 
scale,  telescope,  microscope. 

Directions.  — \a)  Place  the  telescope  at  about  the 
level  of  the  eye,  and  at  some  distance  from  the  scale. 
Focus  the  telescope  on  the  scale  by  moving  the  eyepiece 

in  and  out.  Open  the  left 
eye  and  try  to  see  the  scale 
with  the  left  eye  at  the  same 
time  that  you  see  the  magni- 
FlG-  259-  fied  image  of  the  scale  with 

the  right  eye.  The  two  should  overlap  (see  Fig.  259). 
Count  on  the  scale  the  number  of  divisions  covered  by 
the  magnified  scale.  Try  five  times  and  take  the  aver- 
age of  your  observations. 

(6)  To  find  the  magnifying  power  of  a  microscope 
focus  the  instrument  upon  a  millimetre  scale  and  hold 
a  metre  stick  35  cm.  from  the  eyepiece  on  or  below 
the  table  of  the  microscope.  Compare  the  overlapping 
images  as  in  (a).  If  the  instrument  has  a  micrometer 
eyepiece  the  value  of  a  division  of  the  scale  in  the 
eyepiece  may  be  determined  by  counting  the  number 
of  divisions  of  the  eyepiece  required  to  cover  one 
millimetre  on  the  millimetre  scale  upon  which  the  mi- 
croscope is  focused, 

^   OF  T  H  £ 

UNIVERSITY 


INDEX   OF   PROPER   NAMES. 


[The  numbers  refer  to  pages.    The  names  are  pronounced  as  English  people 
pronounce  them  who  do  not  know  the  foreign  languages.] 


Ampere  (am'pare),  Andre  Marie; 
1755-1836.  French  physicist. 
lie  was  the  first  to  investigate 
carefully  the  relations  be- 
tween electricity  and  mag- 
netism, 130. 

Archimedes  ( a  r-k  y-m  e  e'd  e  e  s } ; 
201-212  B.C.  Greek  mathe- 
matician and  physicist.  He 
invented  numerous  contriv- 
ances to  assist  his  country- 
men in  the  defence  of  Syra- 
cuse against  the  Komans,  47, 
53,  304. 

Becquerel  ( beck-rel' ) ,  Henri; 
French  chemist,  260. 

Boyle,  Robert;  1627-1691.  An 
English  physicist  whose  life 
was  devoted  to  patient  inves- 
tigation .  in  physical  science. 
He  was  particularly  inter- 
ested in  the  study  of  the  air, 
53. 

Crookes,  William;  1832  -  — . 
English  physicist  and  chem- 
ist. His  principal  researches 


in  physics  are  connected  with 
the  molecular  properties  of 
gases.  He  gave  the  name 
"  radiant  matter'1  to  gas  in 
an  extreme  state  of  rarefac- 
tion, 117,  259. 

Daniell  (dan-yel'),  John  Freder- 
ick; 1790-1845.  Author  of 
a  work  on  meteorology.  In- 
ventor of  the  constant  bat- 
tery, 124. 

Fahrenheit  (far'en-hite),  Gabriel 
Daniel.  Physician  and  phys- 
icist of  Dantzig,  65,  66. 

Faraday  ( f air'a-day ) ,  Michael; 
1791-1867.  Renowned  Eng 
lish  chemist  and  physicist. 
He  made  the  first  electric  mo- 
tor and  discovered  the  laws  of 
the  induction  of  currents,  the 
laws  of  electrolytic  action, 
the  identity  of  different  sorts 
of  electrification,  the  different 
inductive  capacity  of  differ- 
ent substances,  the  equality 
of  positive  and  negative 


357 


358 


INDEX  OF  PROPER  NAMES. 


charges.  He  established  all 
of  these  laws  by  numerous 
accurate  experiments  and  laid 
the  foundation  for  the  work 
of  Maxwell,  Hertz,  and  many 
others,  3,  115,  257. 

Franklin,  Benjamin;  1706-1790. 
American  journalist,  diplo- 
mat, and  philosopher.  He 
demonstrated  the  identity  of 
lightning  and  the  electric  dis- 
charge, 113. 

Fraunhofer,  Joseph  von;  1787- 
1826.  Bavarian  optician.  In- 
ventor of  the  spectroscope 
and  first  to  explain  the  dark 
lines  of  the  solar  spectrum. 
First  to  make  and  study  dif- 
fraction gratings,  by  means 
of  which  he  measured  the 
wave  lengths  of  light,  225. 

Galileo  (gal-i-lee'o);  1564-1642. 
One  of  the  earliest  and  great- 
est experimental  p  h  i  1  o  s  o- 
phers.  He  was  professor  at 
Pisa,  where  he  performed  his 
famous  experiments  upon 
falling  bodies  from  the  leaning 
tower,  and  later  at  Florence 
and  Padua.  He  constructed 
the  first  thermometer.  On 
hearing  that  a  German  oculist 
had  made  a  telescope  he  at 
once  constructed  one  on  a 
plan  of  his  own  and  turned 
it  on  the  planets.  He  discov- 


ered the  satellites  of  Jupiter 
and  ushered  in  a  new  era 
in  astronomy  as  well  as  in 
physics.  The  laws  of  motion 
afterwards  formally  stated  by 
Newton  were  first  discovered 
and  taught  by  Galileo,  195. 

Galvani  (gal-vah'nee),  Luigi; 
1703-1785.  Italian  physician 
and  physiologist.  His  discov- 
ery of  the  physiological  ef- 
fects of  electricity  led  to  the 
invention  of  the  battery  by 
Volta  and  stimulated  research 
in  this  hitherto  neglected 
field,  120. 

Geissler,  H  e  i  n  r  i  c  h ;  1814-1879. 
German  glass-blower.  Inven- 
tor of  a  mercurial  air  pump 
and  maker  of  vacuum  tubes, 
117. 

Gilbert,  William;  1540-1603. 
English  physician  and  physi- 
cist. He  made  the  first  care- 
ful study  of  static  electricity 
and  of  magnets,  96. 

Guericke  (ger'ik-e),  Otto  von; 
1602-1686.  German  natural 
philosopher.  He  invented 
the  air  pump  and  constructed 
the  "Magdeburg  hemi- 
spheres," 181. 

Helmholtz,  Herman  Ludwig  Fer- 
dinand; 1848-1894.  Eminent 
German  physiologist  and 
physicist.  His  work  in  all 


INDEX  OF  PKOPEB   NAMES. 


359 


matters  connected  with  the 
sensations  of  sound  and  light 
is  of  the  highest  authority, 
243. 

Heron  or  Hero;  3d  century  B.C. 
Celebrated  Alexandrian  math- 
ematician and  philosopher, 
181. 

Hertz  (e  as  in  very),  Heinrich; 
1857-1894.  German  physicist 
famous  for  his  experiments 
with  electrical  waves.  lie 
had  won  world-wide,  recogni- 
tion when  lie  died  at  the  early 
age  of  37.  The  lines  of  in- 
vestigation opened  up  by  him 
are  being  followed  o  u  t  b  y 
many  investigators  to-day 
with  most  interesting  results, 
258. 

Jolly, .  German  physicist, 

289,  312. 

Joly,  John;  1857.  Irish  scien- 
tist, 351. 

Joule  (jool),  James  Prescott; 
1818-1889.  English  physicist 
noted  for  his  researches  in 
heat,  164. 

Kundt  (koont);  August;  1839 
— .     German  physicist,  217. 

Lenard,    Philipp;    1860 . 

Formerly  assistant  to  Hertz 
at  Bonn,  Germany,  where  he 
made  his  remarkable  series  of 
researches  on  the  electric 


discharge  in  gases.  Now 
Professor  of  Physics  at  Kiel, 
259. 

Marconi  (mar-co'ni),  Guglielmo; 

1875 .  Italian  inventor, 

258. 

Maxwell,  James  Clerk;  1831- 
1879.  Celebrated  Scotch 
physicist.  His  mathematical 
treatises  on  heat  and  elec- 
tricity are  classic,  257,  258. 

Morse,  Samuel  Finley  Breese; 
1791-1872.  American  artist 
and  inventor,  137. 

Newton,  Sir  Isaac;  1642-1727. 
The  greatest  of  natural  phi- 
losophers. His  law  of  univer- 
sal gravitation  is  a  notable 
example  of  the  application 
of  mathematics  to  the  study 
of  physical  problems.  The 
law  of  refraction  grew  out  of 
his  attempts  to  perfect  the 
telescope,  13,  32,  33,  243,  244. 

Oersted  (erst'ed),  Hans  Chris- 
tian; 1777-1851.  Professor  of 
Physics  in  the  University  of 
Copenhagen.  His  discovery 
of  the  magnetic  effect  of  elec- 
tric currents  marked  a  new 
era  in  electrical  studies,  122. 

Ohm  (ome),  Georg  Simon;  1781- 
1854.  Born  at  Erlangen, 
Germany,  and  educated  at 
the  university  there.  He  be- 


360 


INDEX  OF  FBOPER   NAMES. 


came  Professor  of  Experimen- 
tal Physics  at  Munich.  His 
most  important  contribution 
to  science,  the  value  of  which 
can  hardly  be  overestimated, 
is  his  paper  on  the  "Gal- 
vanic Cell  Mathematically 
Treated,1'  129. 

Roemer  (ray'mer),  Ole;  1644- 
1710.  Danish  astronomer, 
256. 

Roentgen  (rent'gen,  g  as  in  get), 
William  Konrad.  B  o  r  n  in 
Holland,  1845;  now  Professor 
of  Physics  at  Wtirtzburg, 
Germany.  He  has  made  nu- 
merous contributions  in  heat 
and  optics,  117,  259. 

Ruhmkorff  (room'korf),  Hein- 
rich  Daniel;  1803-1877.  Ger- 
man-French mechanician,  144. 

Rumf ord,  Count  (Benjamin 
Thomson);  1753-1814.  Born 
in  America,  served  in  the 
British  and  Bavarian  armies. 
Noted  for  his  work  in  heat, 
350. 

Toepler  (tep'ler),  108. 

Volta(vol'ta),  Allesandro;  1745- 
1827.  Professor  of  Physics 
at  Pavia,  Italy.  He  invented 
the  electrophorus,  the  con- 


denser, and  the  absolute  elec- 
trometer, but  is  best  known 
by  his  invention  of  the  bat- 
tery. He  made  careful  exper- 
iments in  gas  analysis  and 
discovered  independently  the 
law  of  Charles,  102,  120. 

Watt,  James;  1736-1819.  The 
inventor  of  the  steam  engine. 
He  was  instrument  maker  in 
Glasgow  College,  where  Pro- 
fessor Black,  the  discoverer 
of  latent  heat,  directed  his 
attention  to  the  study  of  the 
steam  engine.  He  measured 
the  work  done  by  his  engine 
and  denned  the  horse-power, 
181. 

Welsbach  (wels'bock),  133. 

Wheatstone  (wheet'stone),  Sir 
Charles;  1802-1875.  English 
physicist  and  inventor,  best 
known  for  his  study  of  elec- 
tric currents  and  his  inven- 
tions in  telegraphy,  343. 

Young,  Thomas;  1773-1829. 
Discovered  the  law  of  inter- 
ference of  light,  which  went 
far  to  establish  the  undtila- 
tory  theory,  and  suggested 
the  theory  of  color  sensa- 
tion afterwards  developed 
by  Helmholtz,  322. 


GENERAL   INDEX; 


Absolute,  scale  of  temperature, 
69;  zero,  68. 

Absorption,  colors  due  to,  247. 
Acceleration,  10;  of  gravity,  313. 
Adhesion,  35,  41. 

Advantage,  mechanical,  172;  of 
inclined  plane,  178;  of  pulley, 
174;  of  screw,  179;  of  wedge, 
178. 

Air,  elasticity  of,  35;  com- 
pressed, for  transmission  of 
power,  188;  pressure  of,  49; 
resistance  of,  25. 

Air  pump,  54. 

Air  thermometer,  67. 

Alternating,  current,  148;  dy- 
namo, 148. 

Altitude,  measured  with  barom- 
eter, 318. 

Ampere,  unit  of  current,  130. 

Amplitude,  of  vibration,  193;  re- 
lated to  loudness,  212. 

Antinodes,  in  vibrating  bodies, 
202. 

Arc  lamp,  132. 
*  The  references  are  to  pages. 


Archimedes'  principle,  47. 
Atoms,  63. 

Attraction,  of  cohesion,  35,  41; 
of  electric  charges,  94;  of 
electric  currents,  135 ;  of  grav- 
itation, 33 ;  of  magnets,  86, 94. 

Aurora  borealis,  117. 

Balance,  spring,  289;  lever,  31, 
290,  292. 

Barometer,  50,  316;  altitude 
measured  with,  318;  storms 
indicated  by,  49. 

Battery,  electric,  120,  123. 
Beats,  in  music,  210,  212. 

Bells,  electric,  139 ;  vibration  of, 
209. 

Boiling  point,  of  liquids,  74;  of 
water,  329. 

Boyle's  law,  53,  320. 
Breaking  stress,  326. 
Bridge,  Wheatstone's,  340. 
Bubbles,  form  of,  45. 
Buoyant  force,  of  fluids,  47. 
Calipers,  276,  277. 


361 


362 


GENERAL  INDEX. 


Calorie,  unit  of  heat,  70. 
Calorimetry,  69,  333. 
Camera,  239,  240. 

Candle  power,  256 ;  measurement 
of,  350. 

Capacity,  electrostatic,  110;  for 
heat,  70,  333;  specific  induc- 
tive, 99. 

Capillarity,  42,  320. 

Capillary  phenomena,  41;  con- 
stant table  of,  322. 

Cathode  rays,  116,  258. 
Cell,  voltaic,  120. 

Centigrade  scale  of  temperature, 
65. 

Centre,  of  mass,  28 ;  of  gravity, 
28;  optical,  of  a  lens,  238. 

Centrifugal  force,  14. 

Changes,  physical,  62;  chemical, 
62. 

Clothing,  79. 

Coefficient,  of  expansion,  332; 
table  of,  333;  of  friction,  328; 
temperature  of  resistance,  343. 

Cohesion,  35,  41. 
Cold,  artificial,  66. 

Color,  by  absorption,  247;  by 
diffraction,  251;  by  refraction, 
242 ;  sensation  of,  242 ;  mixture 
of  sensations  of,  244-246. 


Colors,  by  thin  plates,  251 ;  com- 
plementary, 246 ;  interference, 
249,  251 ;  mixture  of,  244-246. 

Combustion,  a  source  of  energy, 
168,  183. 

Commutator,  of  dynamo,  141. 
Compass,  magnetic,  94. 

Composition,  of  force,  14,  17;  of 
velocities,  17. 

Compressed  air,  for  transmission 
of  power,  188. 

Condenser,  electric,  112. 

Conduction,  of  heat,  76 ;  of  elec- 
tricity, 96,  129. 

Conductors,  electric,  96,  129. 
Conservation,  of  energy,  161. 
Consonance,  212. 

Constant,  capillary,  320;  of  grav- 
itation, 33. 

Contact,  electric  charge  by,  120. 
Convection,  of  heat,  77. 
Crookes'  tube,  258. 
Crystals,  71. 

Current,  electric,  111,  120,  129, 
130;  alternating,  148;  attrac- 
tions and  repulsions  of,  135; 
distribution  of  power  by,  188; 
mechanical  effect  of,  126; 
Ohm's  law  for,  130;  produced 
by  batteries,  120;  by  heat,  134; 
by  motion,  142;  steady,  147; 


GENERAL  INDEX. 


363 


producing  chemical  effects, 
126;  producing  heat,  128;  pro- 
ducing magnetic  effects,  122. 

Curvature,  of  lenses,  287. 
Declination,  magnetic,  94. 

Degree,  Centigrade,  66;  Fahren- 
heit, 66. 

Density,  by  Archimedes'  prin- 
ciple, 304,  305,  308;  linear,  300; 
maximum  of  water,  71;  of 
.  liquids,  302-308;  of  regular 
solids,  301;  surface,  300; 
tables  of,  3$9. 

Dew  point,  336. 
Diagonal  scale,  274. 
Diffraction  spectrum,  250. 

Diffusion,  of  fluids,  55 ;  of  gases, 
56;  of  liquids,  58. 

Discharge,  electric,  from  points, 
106;  in  a  vacuum,  116,  258. 

Dissonance,  212. 
Distillation,  74. 
Dynamo,  electric,  145. 
Dyne,  22. 

Earth,  a  magnet,  94. 
Ebullition,  73. 

Efficiency,  of  machines,  172;  of 
pulley,  328. 

Elastic  forces,  37. 

Elasticity,  34,  322;  explained,  37; 
modulus  of,  322 ;  of  air,  35. 


Electric  attractions,  94. 
Electric  battery,  120,  123. 
Electric  bell,    139. 

Electric  charge,  94;  by  contact, 
120;  nature  of,  100;  distribu- 
tion of,  103. 

Electric  current,  111,  120,  129, 
130;  alternating,  148;  attrac- 
tions and  repulsions  of,  135; 
chemical  effect  of,  126;  effect 
in  producing  heat,  128;  for 
distribution  of  energy,  188; 
magnetic  effect  of,  126;  me- 
chanical effect  of,  126;  Ohm's 
law  for  intensity,  130;  pro- 
duced, by  batteries,  120;  by 
heat,  134;  by  motion,  142; 
unit  of,  130. 

Electric  discharge,  in  air,  104;  in 
rarefied  gases,  117;  in  tubes 
(Geissler's  and  Crookes'),  117. 

Electric,  heating,  133;  lighting, 
131;  motor,  140;  potential, 
130,  345;  repulsions,  94;  resist- 
ance, 128,  130,  339,  342,  344; 
units,  130;  welding,  133. 

Electrical  machines,  107-110. 

Electricity,  84;  and  light,  256; 
galvanic,  120 ;  measurement 
of,  339-345. 

Electrolysis,  126. 
Electro-magnetic  telegraph,  137. 
Electro-magnets,  136. 


364 


GENERAL  INDEX. 


Electrophorus,  102. 
Electroplating,  127. 

Electroscope,  pith-ball,  95;  gold- 
leaf,  114. 

Electrotyping,  128. 

Energy,  157,  159;  and  life,  166; 
conservation  of,  160,  161;  dis- 
tribution of,  187;  equivalents 
of  different  forms,  162 ;  forms 
of,  165;  kinetic,  160, 162;  illus- 
trated in  pendulum,  194;  of 
falling  bodies,  163;  potential, 
160,  162;  illustrated  in  pendu- 
lum, 194;  radiant,  see  radiant 
energy ;  sources  of  useful,  167 ; 
storage  of,  186;  sun  a  source 
of,  167,  260;  transference  of, 
187;  units  of,  162. 

Engine,  gas,  183,  184;  steam, 
Hero's,  181;  Watts',  181,  182. 

Equilibrium,  27-29;  kinds  of,  27. 

Equivalent,  Joule's,  of  work  and 
heat,  164. 

Equivalents,  metric  and  English, 

20,  268,  288. 

Erg,  unit  of  work,  158. 
Errors,  of  measurement,  266. 

Estimation,  of  tenths,  275,  279; 
of  time,  314. 

Ether,  92,  260;  waves  in,  see  ra- 
diant energy. 


Evaporation,  73;  a  cooling  proc- 
ess, 75. 

Exercises,  1-12,  in  matter  and 
motion,  23,  24;  13-31,  in  bal- 
ancing forces,  59,  60;  32-51, 
in  beat,  82,  83;  52-69,  in  mag- 
netism and  static  electricity, 
117-119;  70-88,  in  electric  cur- 
rents, 151-156;  89-101,  in 
work  and  machines,  189-191 ; 
102-107,  in  vibrations  and 
waves,  205,  206;  108-116,  in 
sound,  224,  225;  117-130,  in 
light,  260. 

Exercises,  laboratory,  131-144, 
in  length,  269-287;  145-165, 
in  mass  and  density,  288-309 ; 
166-173,  in  time,  311-315;  174- 
183,  in  force,  316-328;  184- 
191,  in  heat,  329-338;  192-197, 
in  electricity,  339-345;  198- 
200,  in  sound,  346-349;  201- 
206,  in  light,  350-356. 

Expansion,  at  solidification,  71; 
by  heat,  64;  coefficient  of,  64, 
332;  table  of,  333;  of  gases, 
64;  of  liquids,  64;  of  mer- 
cury, 67;  of  water,  64;  un- 
equal, of  metals,  68. 

Eye,  240;  estimation  of  length, 

282. 

Fahrenheit  scale  of  temperature, 
65. 

Faraday's  ice-pail  experiment, 
115. 


GENERAL   INDEX. 


365 


Falling  bodies,  2,  10,  11;  energy 
of,  163. 

Field,  magnetic,  88;  nature  of, 
92;  of  force,  electrostatic,  98; 
magnetic,  about  a  current, 
123. 

Floating  bodies,  46 ;  Archimedes' 
law  of,  46. 

Fluids,  38;  in  contact,  55. 
Fluoroscope,  258. 

Focus,  of  mirror,  234;  of  lens, 
236. 

Foot-pound,  J58. 

Force,  centrifugal,  14;    denned, 
8;  lines  of,  93,  98,  123;  meas-  . 
urement  of,  316-328;  moment 
of,  30;  nature  of,  9. 

Forces,  25;  composition  of,  14; 
elastic,  37;  resolution  of,  14; 
triangle  of,  16. 

Fraunhofer's  lines,  254. 

Freezing  mixture,  66. 

Friction,  26;  coefficient  of,  327. 

Fundamental,  tone,   213;    units, 

20,  265. 

Fusion,  71;  latent  heat  of,  76. 
Galvanometer,  123. 
Galvanoscope,  123. 

Gases  and  liquids,  how  different, 
48;  Archimedes'  law  applies 
to,  53;  expansion  of,  69;  in 


closed    vessels,   53;    in  open 
vessels,  48;  pressure  of,  53. 

Gaseous  state,  36. 
Gram,  denned,  21,  288. 
Grating,  diffraction,  250,  252. 

Gravitation,  32;  constant  of,  33; 
Newton's  law  of,  33. 

Gravity,  9 ;  acceleration  of,  meas- 
ured by  pendulum,  313;  cen- 
tre of,  28;  force  of,  in  dynes, 
22,  313;  opposing  motion,  25. 

Harmonic  motion,  193. 
Harmonics,  overtones,  220 
Harmony,  212. 

Heat,  36;  amount  of,  69;  capac- 
ity, 70,  333;  conduction,  76; 
convection,  77;  effect  of,  62; 
on  a  magnet,  93;  engines, 
180-184;  expansion  due  to, 
62;  tables  of,  333;  latent,  of 
evaporation,  75 ;  of  fusion,  76 ; 
measurement  of,  329 ;  mechan- 
ical equivalent,  164;  nature 
of,  61;  produced  by  electric 
current,  128;  quantity  of,  69; 
radiant,  see  radiant  energy; 
sensation  of,  64;  specific,  70, 
333;  table  of,  336;  units  of, 
69. 

Heating,  electric,  133;  of  houses, 

79. 

Horse  power,  unit  of  work,  184. 


366 


GENERAL  INDEX. 


Hue,  244. 

Humidity,   absolute,   336;    rela- 
tive,  336;    table   of  relative, 


Hygrometry,  336. 
Hypothesis,  2. 

Ice-pail  experiment,  Faraday's, 
115. 

Images,  by  reflection,  229-232; 
by  refraction,  235;  by  small 
openings,  228;  real,  230;  vir- 
tual, 241. 

Incandescent  lamp,  132. 
Inclined  plane,  24,  178. 

Index  of  refraction,  204,  351; 
table,  354. 

Induction,  coil,  144;  electric,  of 
currents,  143;  electrostatic, 
98;  magnetic,  explained,  90. 

Inductive  capacity,  99. 
Inertia,  13. 

Instruments,  optical,  239-242. 
Insulators,  electric,  96. 

Intensity,  of  light,  255;  of  elec- 
tric current,  130;    of  field  of 
,  force,  99 ;  of  sound,  212. 

Interference,  of  light  waves,  249 ; 
of  sound  waves,  215,  217,  218, 
219;  of  waves,  201,  218,  219; 
spectrum,  250. 

Intervals,  musical,  211. 


Jolly's  balance,  289. 
Joule,  unit  of  work,  184. 

Joule's  equivalent  of  heat  and 
work,  164/ 

Kinetic  energy,  160,  162,  194. 
Kundt's  tube,  217,  347. 

Lamp,  arc,  132;  incandescent, 
132. 

Latent  heat,  of  fusion,  76 ;  of 
vaporization,  75. 

Law,  Archimedes1,  47;  Boyle's, 
of  gases,  53;  defined,  2;  of 
Boyle,  320;  of  Charles,  68;  of 
conservation  of  energy,  161; 
of  electrostatic  attraction,  98; 
of  falling  bodies,  2 ;  of  inten- 
sity of  light,  256,  350;  of  mag- 
netic force,  86;  of  motion, 
13-19. 

Length,  eye  estimation  of,  282; 
measurement  of,  268-287; 
units  of,  20,  268. 

Lenses,  235-239;  curvature  of, 
286 ;  focal  length  of,  354. 

Lever,  170;  balance,  31;  princi- 
ple of,  30. 

Leyden  jar,  113. 

Light,  and  elasticity,  256;  defini- 
tions, 226;  diffraction  of,  251; 
intensity  of,  350;  interference 
of,  251,  252;  measurement, 
350-356;  photometry,  350; 


HE  VIRAL   INDEX. 


367 


rectilinear  propagation  of, 
227,  228;  reflection  of,  229- 
234;  refraction  of,  234,  241, 
351. 

Lighting,  electric,  131. 
Lightning,  105,  106. 

Lines  of  force,  electrostatic,  99, 
101;  in  the  dynamo,  146;  mag- 
netic, 93,  135. 

Liquid  state,  36. 

Liquids,  boiling  point  of,  73; 
density  of,  302-308;  equilib- 
rium of,  39;  evaporation  of, 
73;  expansion  of,  64;  in  closed 
vessels,  47;  in  communicating 
vessels,  40;  in  open  vessels, 
39-47;  laws  of  pressure,  40. 

Litre,  21,  269. 

Lodestone,  85. 

Loops,  in  wave  motion,  see  an- 
tinodes. 

Machines,  157,  169 ;  efficiency  of, 
172. 

Magic  lantern,  241. 

Magnetic,  attraction  and  repul- 
sion, 86,  94;  compass,  94;  ef- 
fects of  electric  current,  126; 
field,  88;  force,  law  of,  87; 
induction,  90;  lines  of  force, 
88,  93,  99;  needle,  94;  poles, 
85;  substances,  91. 

Magnetism,  84;  destroyed  by 
heat,  93;  induced,  87;  molec- 
ular, 91;  of  earth,  94. 


Magnets,  85. 
Magnification,  356. 

Major,  scale,  221;  triad,  221, 
222. 

Manometer,  319. 

Mass,  11,  12;  centre  of,  28; 
measurement  of,  288-309; 
units  of,  20,  288. 

Matter,  defined,  4;  some  prop- 
erties of,  34. 

Measurement,  errors  of,  266;  ex- 
ercises in,  see  exercises;  phys- 
ical, 265;  units  of,  20. 

Mechanical  advantage,  172 ;  equiv- 
alent, 164. 

Melting  point,  of  a  solid,  71, 331. 

Mercury,  in  capillary  tubes,  42. 

Metre,  20,  268. 

Micrometer,  283. 

Microscope,  239. 

Mirrors,  229-234. 

Modulus  of  elasticity,  322. 

Molecules,  35;  motion  of,  in 
heat,  62. 

Moment,  of  force,  30;  applied 
to  lever,  171. 

Momentum,  11,  12. 

Motion,  defined,  4;  harmonic, 
193;  Newton's  laws  of,  13- 
19;  rate  of,  10;  rotary,  7; 


368 


GENERAL  INDEX. 


translatory,    5;    vibratory,    7, 
192;  wave,  7,  197. 

Motor,  electric,  140. 
Musical  scales,  213,  220,  221. 

Nodes,  in  bells,  210;  in  organ 
pipes,  216 ;  in  vibrating  bodies, 
202,  217,  347. 

Note-book,  for  exercises,  267. 
Ohm,  unit  of  resistance,  130. 
Ohm's  law,  129 

Optical,  centre  of  a  lens,  238 ;  in- 
struments, 239-242. 

Organ  pipes,  216. 
Oscillation  of  pendulum,  311. 
Osmose,  58. 
Overtones,  213,  220. 

Parallelogram  of  forces,  see  com- 
position of  forces. 

Pendulum,  194;  compensating 
period  of,  195;  simple,  196; 
time  of  oscillation,  311. 

Penumbra,  228. 

Period,  of  vibration,  193;  of  pen- 
dulum, 195. 

Perpetual  motion,  185. 
Photometry,  256,  350. 

Physical,  changes,  62;  measure- 
ment, 265. 

Physics,  founded  on  measure- 
ment, 265;  place  among  sci- 


ences,   1;    place   in   practical 
life,  3. 

Pigments,  mixture  of,  248. 
Pile  driver,  159,  195. 
Pitch,  in  sound,  211. 
Plates,  vibration  of,  209. 

Polarization,  of  electric  batter- 
ies, 122. 

Poles,  of  a  magnet,  85. 
Polygon,  of  forces,  17. 
Pores,  35. 

Potential,  measurement  of,  345; 
electrostatic,  110;  unit  of,  130. 

Power,  and  weight,  old  terms 
for  force,  171;  rate  of  doing 
work,  184;  transmission  of, 
187;  Boyle's  law  of,  53,320. 

Pressure,  of  air,  316;  of  fluids, 
319. 

Properties,  of  matter,  34. 

Pulley,  174-177;  efficiency  of, 
328. 

Pulse,  to  count,  314. 

Pump,  for  air,  54;  for  liquids, 
51. 

Quality,  of  musical  tones,  212. 
Quantity,  of  heat,  69. 

Radiant  energy,  61,  165,  256,  257, 
259;  absorption  of  producing 
color,  247;  cause  of  sensation 


GENERAL  INDEX. 


369 


of  light,  226;  law  of  intensity 
of,  255;  of  sun,  168,  169;  stor- 
age of,  by  plants,  187;  trans- 
formed into  heat,  61;  velocity 
of,  254;  wasted,  186. 

Radiation,  new  forms  of,  258. 

Rainbow,  243. 

Ray,  227. 

Reaction,  19. 

Record  of  experiments,  267. 

Reflection,  law  of,  202 ;  of  light, 
229-234;  of  waves,  200-203. 

Refraction,  color  produced  by, 
243;  index  of,  204,  351;  of 
waves,  204. 

Relay,  telegraphic,  138. 

Resistance,  box,  339;  by  substi- 
tution, 33D;  electric,  128; 
measurement  of,  339;  specific, 
342;  table  of,  344;  unit  of, 
130. 

Resolution,  of  forces,  14,  17;  of 
velocities,  17. 

Resonance,  214;  of  air  columns, 
215,  346;  of  organ  pipes,  216. 

Respiration,  to  count,  314. 
Rods,  vibration  of,  208. 

Rotary  motion,  7;  of  the  earth, 
14. 

Ruhmkorff' s  coil,  144. 


Scale,  absolute,  of  temperature, 
69;  chromatic,  224;  diagonal, 
274;  major,  221;  tempered, 
224. 

Scales,  musical,  213,  219,  220. 
Screw,  179;  micrometer,  283. 
Shades,  of  color,  245. 
Shadows,  227.      ./""" 
Siphon,  52. 
Siren,  211. 
Solenoid,  136. 
Solid  state,  36. 

Sound,  and  noise,  207;  loudness 
of,  212;  measurement  of,  346- 
349;  nature  of,  207;  pitch  of, 
211;  quality  of,  212;  velocity 
in  air,  216,  346;  velocity  in 
solids,  217,  347. 

Sounder,  telegraphic,  138. 
Space  telegraphy,  257. 
Spark  coil,  Ruhmkorff  s,  144. 

Specific,  gravity,  see  density; 
heat,  70,  333;  inductive  ca- 
pacity, 99;  resistance,  342. 

Spectra,  classes  of,  253. 
Spectroscope,  251,  252. 
Spectrum  analysis,  253. 

Spectrum,  by  diffraction,  250; 
by  refraction,  242;  of  sun, 
254. 


370 


GENERAL  INDEX. 


Speed,  10. 
Spherometer,  285. 
Stability,  conditions  of,  29. 

State  of  bodies,  change  of,  due  to 
heat,  62;  gaseous,  48;  liquid, 
36;  solid,  36. 

Steelyards,  32,  292. 

Stereopticon,  241. 

Still,  75. 

Storms,  indicated  by  barometer, 

49. 

Strain,  34;  electrical,  104. 
Strength  of  a  wire,  326. 

Stress,  34;  electrical,  101;  break- 
ing, 326. 

Strings,  vibrations  of,  219. 

Sun,  a  source  of  energy,  167, 
260;  see  also  radiant  energy. 

Surface  tension,  43-46;  see  also 
capillarity. 

Tables,  of  breaking  stress,  326; 
of  capillary  constant,  322;  of 
coefficient  of  expansion,  333;  \ 
of  specific  heat,  336;  of  diam- 
eters of  wires,  285;  of  densi- 
ties, 309;  of  elasticity,  326;  of 
index  of  refraction,  354;  of 
lengths,  270;  of  resistance, 
344;  of  velocity  of  sound,  347; 
of  weights,  288. 

Telegraph,  electro-magnetic,  137 ; 
Morse's  printing,  137. 


Telegraphy,  without  wires,  257. 

Telephone,  156;  used  in  measur- 
ing resistance,  344. 

Telescope,  magnification  of,  356. 

Temperature,  63;  measurement 
of,  329;  of  the  body,  77,  82, 
331 ;  scales  of,  65 ;  unit  of,  65. 

Tension,  surface,  43-46. 
Tenths,  estimation  of,  269,  275. 
Theory,  2;  atomic,  2. 
Thermo-electric  current,  134. 
Thermo-electric  generators,  134. 

Thermometer,  65;  air,  67;  cali- 
bration of,  329. 

Thermometry,  65. 
Thermostat,  68. 

Time,  measurement,  310-314; 
units  of,  20,  310. 

Tint,  244. 

Tones,  fundamental,  213;  par- 
tial, 213. 

Transformer,  for  alternating  cur- 
rents, 150. 

Translation,  motion  of,  5. 
Triangle  of  forces,  16. 

Tuning  fork,  209;  vibration 
number  of,  349. 

Units,  C.  G.  S.  system,  21;  de- 
rived, 21;  electric,  130;  fun- 


GENERAL  INDEX. 


371 


dameiital,  20,  265;  of  accelera- 
tion, 21;  of  force,  21;  of  heat, 
69;  of  length,  20;  of  magnetic 
force,  87;  of  momentum,  21; 
of  time,  20;  of  velocity,  21; 
of  wave  lengths,  243;  of  work, 
158,  184. 

Unit  pole,  87. 

Vaporization,  72;  latent  heat  of, 
75. 

Velocity,  10;  of  light,  254,  351; 
of  sound,  216,  217,  346;  table 
of,  347. 

Velocities,  composition  of,  17; 
resolution  of,  17. 

Ventilation,  65;  of  houses,  79. 
Vernier,  278;  gauge,  280. 

Vibration,  number  of  tuning 
fork,  349;  of  bells,  209;  of 
pendulum,  311;  of  plates,  209; 
of  rods,  208;  of  strings,  219. 

Vibratory  motion,  7,  192. 
Vision,  errors  of,  241. 
Volt,  unit  of  potential,  130. 

Volume,  by  Archimedes'  princi- 
ple, 47,  304;  by  displacement, 
303;  of  irregular  solids,  47; 
tables  of,  309. 

Water  wheels,  185. 


Watt,  unit  of  power,  184. 

Wave,  length,  unit  of,  243;  mo- 
tion, 7. 

Waves,  197;  direction  of,  199; 
electrical,  257;  longitudinal, 
200;  reflection  of,  200-203;  in 
rods,  208;  stationary,  202. 

Weather,  78. 
Wedge,  178. 

Weighing,  by  swings,  296; 
double,  295;  with  balance, 

289. 

Weight,  in  vacuum,  53. 

Weights,  directions  for  making, 
301;  table  of,  288. 

Welding,  electric,  133. 
Wheatstone's  bridge,  340. 

Winds,  78;  a  source  of  energy, 
168. 

Work,  157;  rate  of  doing,  184; 
units  of,  158;  why  we  do,  166; 
see  also  energy  and  radiant 
energy. 

X-rays,  258. 

Zero,  absolute,  68;  Centigrade, 
66,  329;  error  of,  265;  Fahren- 
heit, 66;  of  thermometer, 
329. 


LOAN  DEPT. 


-50m-8,'61 


